251 Differential Equations Terms and Definitions | Mathematics Board Exam Review

INTRODUCTION

Differential equations are the language of engineering analysis. Every time an engineer models a vibrating machine, analyzes a circuit under transient conditions, computes the temperature distribution in a structural member, or predicts the behavior of a control system, the underlying mathematics is a differential equation. For Mathematics engineering licensure examinees, differential equations is one of the most applied and most heavily weighted topics in the engineering mathematics portion of the PRC board exams. The subject bridges pure calculus with real engineering systems, and the ability to recognize, set up, and solve differential equations is a skill that carries points across all major engineering disciplines.

What makes this subject particularly demanding is the variety of equation types and solution methods. A first order separable equation is solved in a completely different way from a second order equation with constant coefficients, which in turn requires a different approach from an equation solved by the Laplace transform. Add to this the range of applications from mechanical vibrations and electrical circuits to heat conduction and population dynamics and the subject demands both breadth and depth. Reviewees who approach differential equations as a collection of disconnected formulas tend to struggle. Those who understand the structure of each equation type and the logic behind each solution method consistently perform better under exam conditions.

This reference list compiles 251 terms and definitions drawn from the full scope of differential equations as tested in Mathematics engineering board examinations. The list covers first order equations including linear, separable, exact, Bernoulli, and homogeneous types; second order equations with constant and variable coefficients; methods including undetermined coefficients, variation of parameters, and the Laplace transform; systems of differential equations and their matrix formulation; power series and Frobenius solutions; partial differential equations including the heat, wave, and Laplace equations; numerical methods from Euler to Runge-Kutta; and the qualitative theory of stability and phase plane analysis. Every definition is written to be directly useful in board exam preparation.

Use this list strategically. Focus first on the foundational terms that appear in every major topic: order, linearity, the general solution, initial conditions, and the superposition principle. Then work through the solution methods in order of frequency on the PRC exams: first order linear equations and their integrating factors, second order constant coefficient equations and the characteristic equation, the Laplace transform method, and the method of undetermined coefficients. Application terms such as mixing problems, RLC circuits, spring-mass systems, Newton’s law of cooling, and resonance are almost always tested as word problems, so understanding these deeply is essential. Return to this list often throughout your review and pair it with worked examples to reinforce each concept.

The 251 Differential Equations Terms and Definitions

1. Abel’s Theorem

A theorem concerning the Wronskian of solutions to a linear second order differential equation. It states that the Wronskian is either identically zero or never zero on an interval where the coefficients are continuous. This result is used to verify the linear independence of solution sets without computing the Wronskian explicitly at every point.

2. Abel-Liouville Formula

A formula expressing the Wronskian of solutions to a second order linear differential equation in terms of the coefficient of the first derivative. It states that the Wronskian at any point equals the Wronskian at an initial point multiplied by an exponential factor involving the integral of that coefficient. This formula is used to compute the Wronskian without finding the solutions explicitly.

3. Adams-Bashforth Method

A family of explicit multi-step numerical methods for solving initial value problems that use derivative information from several previous steps to advance the solution. Higher-order Adams-Bashforth methods achieve greater accuracy than single-step methods of the same order with fewer function evaluations per step, making them efficient for long-time integration.

4. Adams-Moulton Method

A family of implicit multi-step numerical methods that are used as corrector steps in predictor-corrector schemes, typically paired with Adams-Bashforth predictors. Adams-Moulton methods achieve higher accuracy than explicit methods of the same number of steps and are particularly effective for non-stiff equations where moderate accuracy is needed over long intervals.

5. Adjoint Equation

The differential equation obtained from a given linear differential equation by applying the adjoint operator. The adjoint equation is used in the construction of Green’s functions and in the derivation of self-adjoint forms. It plays an important role in the theory of boundary value problems.

6. Amplitude of Oscillation

The maximum displacement from equilibrium in an oscillatory solution to a differential equation. For an undamped harmonic oscillator, the amplitude is constant and determined by initial conditions. For an underdamped system, the amplitude decays exponentially over time. Amplitude is one of the key parameters extracted from the solution form A times e to the negative bt times cosine of omega t plus phi.

7. Analytic Coefficient

A coefficient function of a differential equation that can be represented by a convergent power series in a neighborhood of a given point. At points where all coefficients are analytic, the differential equation has an ordinary point and the power series method yields two independent analytic solutions. Analytic coefficients are the condition required for the power series method to apply.

8. Analytic Solution

A solution to a differential equation expressed as a closed-form combination of elementary functions or known special functions. Analytic solutions are preferred in engineering because they reveal the behavior of the solution explicitly and allow straightforward computation of values at any point.

9. Annihilator Method

An alternative approach to the method of undetermined coefficients that uses differential operators to annihilate the forcing function. The annihilator of the forcing function is applied to both sides of the equation, raising the order and converting the problem to a homogeneous equation whose general solution contains the particular solution as a subset.

10. Asymptotic Behavior

The long-term behavior of the solution to a differential equation as the independent variable grows without bound. For stable linear systems, solutions decay to zero or approach a steady state. For unstable systems, solutions grow exponentially. Determining asymptotic behavior without solving the equation completely is a key goal of qualitative analysis.

11. Asymptotic Stability

A property of an equilibrium point where nearby solutions not only remain close but also converge to the equilibrium as time increases without bound. Asymptotic stability is the strongest and most practically useful form of stability in engineering system analysis.

12. Attractor

A set in the phase space toward which nearby trajectories converge as time increases. A stable equilibrium point is a point attractor. A stable limit cycle is a periodic attractor. Attractors characterize the long-term behavior of dissipative systems and are central to the analysis of nonlinear dynamics in engineering.

13. Autonomous Differential Equation

A differential equation in which the independent variable does not appear explicitly, meaning the rate of change depends only on the current state of the system. Autonomous equations are important in modeling systems where the dynamics are self-contained and time-invariant.

14. Autonomous System

A system of differential equations in which the right-hand side functions do not depend explicitly on the independent variable, only on the dependent variables. The phase portrait of an autonomous system is fixed and trajectories cannot cross. Autonomous systems are used to model time-invariant physical processes including population dynamics and mechanical oscillations.

15. Auxiliary Equation

The algebraic equation obtained by substituting an assumed exponential solution into a linear differential equation with constant coefficients. Also called the characteristic equation, it determines the form of the complementary function. Its roots may be real distinct, real repeated, or complex conjugate pairs.

16. Backward Euler Method

A numerical method for solving differential equations that uses the derivative at the next time step rather than the current one, making it an implicit method. The backward Euler method is unconditionally stable for stiff equations, making it preferred in engineering simulations where explicit methods would require extremely small step sizes.

17. Beats Phenomenon

A physical behavior arising in forced oscillation problems where the driving frequency is close but not equal to the natural frequency of the system. The solution exhibits a pattern of alternating amplification and cancellation described by the product of two sinusoidal terms with slightly different frequencies. It is a direct consequence of the particular solution structure.

18. Bernoulli Differential Equation

A nonlinear first order differential equation of the form where the derivative of y equals a function of x times y plus another function of x times y raised to a power n. The equation is linearized by the substitution v equals y raised to one minus n, converting it into a linear first order equation that can be solved by standard methods.

19. Bessel’s Equation

A second order linear differential equation with variable coefficients that arises when solving problems with cylindrical symmetry such as heat conduction in cylinders and vibrations of circular membranes. Its solutions are called Bessel functions of the first and second kind and are among the most important special functions in engineering mathematics.

20. Bessel Functions

The solutions to Bessel’s equation, named after Friedrich Bessel. Bessel functions of the first kind are finite at the origin, while those of the second kind become infinite there. They appear in problems involving circular or cylindrical geometry and are tabulated for use in engineering design and analysis.

21. Bifurcation

A qualitative change in the behavior of solutions to a differential equation as a parameter crosses a critical value. At the bifurcation point, the number or stability of equilibrium points changes. Bifurcation analysis is used in engineering to identify thresholds where system behavior shifts qualitatively.

22. Boundary Condition

A condition imposed on the solution of a differential equation at the boundary of the domain rather than at an initial point. Boundary conditions are used in boundary value problems where the behavior of the solution is specified at two or more distinct points, such as the ends of a beam or a heat-conducting rod.

23. Boundary Layer

A thin region near a boundary where the solution to a differential equation changes rapidly. Boundary layers arise in singular perturbation problems where a small parameter multiplies the highest derivative. Engineering problems involving viscous flow, heat transfer with high Peclet number, and stiff systems all exhibit boundary layer phenomena.

24. Boundary Value Problem

A differential equation paired with conditions specified at two or more distinct points in the domain. Unlike initial value problems, boundary value problems may have no solution, a unique solution, or infinitely many solutions. They model physical situations such as steady-state heat conduction and beam deflection.

25. Cauchy-Euler Equation

A second order linear differential equation with variable coefficients where each coefficient is a constant times the corresponding power of the independent variable. It is solved by the substitution x equals e to the t, which converts it into a linear equation with constant coefficients that can be solved using the auxiliary equation.

26. Center

An equilibrium point of an autonomous system around which solutions trace closed orbits, indicating periodic behavior. A center is neither stable nor unstable in the Lyapunov sense because nearby trajectories neither converge to nor diverge from the equilibrium. It appears in undamped oscillatory systems.

27. Characteristic Equation

The polynomial equation obtained from a linear differential equation with constant coefficients by replacing each derivative with the corresponding power of the characteristic variable r or lambda. The roots of the characteristic equation determine the form of the general solution to the homogeneous equation.

28. Characteristic Curve

A curve in the domain of a partial differential equation along which information propagates. The method of characteristics converts the partial differential equation into ordinary differential equations along these curves. For hyperbolic equations such as the wave equation, characteristics are real and distinct and carry disturbances through the domain.

29. Characteristic Direction

The direction in the xt-plane along which a partial differential equation reduces to an ordinary differential equation in the method of characteristics. For the wave equation, characteristics have slopes equal to the positive and negative reciprocals of the wave speed and represent the paths along which wave fronts travel.

30. Characteristic Roots

The solutions to the characteristic equation of a linear differential equation with constant coefficients. Real distinct roots yield independent exponential solutions, repeated roots require multiplication by powers of x, and complex conjugate roots produce sinusoidal solutions. The nature of the roots determines the qualitative behavior of the system.

31. Clairaut’s Equation

A first order differential equation of the form y equals x times dy/dx plus a function of dy/dx. Its general solution is a family of straight lines, and it also possesses a singular solution that is the envelope of this family. Clairaut equations appear in geometry and optics problems.

32. Complementary Function

The general solution to the associated homogeneous differential equation, which forms one part of the complete solution to a nonhomogeneous equation. The complementary function contains arbitrary constants equal in number to the order of the equation and represents the natural response of the system.

33. Complete Solution

The full general solution to a nonhomogeneous differential equation, consisting of the complementary function added to any particular solution. The complete solution contains all arbitrary constants and represents the entire family of functions satisfying the differential equation.

34. Complex Exponential Solution

A solution to a linear differential equation with constant coefficients expressed using complex exponentials of the form e raised to a complex power. When characteristic roots are complex conjugates, the complex exponential solutions are combined using Euler’s formula to produce real-valued sinusoidal solutions. Complex exponentials simplify the algebra of oscillatory solutions.

35. Complex Roots

Characteristic roots that occur as conjugate pairs of the form alpha plus and minus i times beta. They arise when the discriminant of the characteristic equation is negative. The corresponding solutions are exponentials with sinusoidal factors: e to the alpha x times cosine of beta x and e to the alpha x times sine of beta x.

36. Conservative System

A differential equation system in which a conserved quantity, such as total energy, remains constant along every solution trajectory. Conservative systems have no attractors and their phase portraits consist entirely of closed orbits. The undamped harmonic oscillator is the standard example of a conservative mechanical system.

37. Constant Coefficient Equation

A linear differential equation in which all coefficients of the dependent variable and its derivatives are constants rather than functions of the independent variable. These equations are solved systematically using the characteristic equation and are the most important class of linear differential equations in engineering applications.

38. Convergence of Numerical Methods

The property of a numerical method for solving differential equations where the numerical solution approaches the exact solution as the step size approaches zero. Convergence is a fundamental requirement for a numerical method to be useful and is related to the concepts of consistency and stability through the Lax equivalence theorem.

39. Convolution

An operation that combines two functions by integrating the product of one function with a reversed and shifted version of the other. Convolution appears in the Laplace transform method as the inverse transform of a product of transforms. It is used to express the particular solution of a linear equation in terms of the Green’s function.

40. Convolution Theorem

The Laplace transform result stating that the transform of the convolution of two functions equals the product of their individual transforms. This theorem allows the particular solution of a linear differential equation to be written as the inverse transform of the product of the transfer function and the transform of the forcing function.

41. Coupled Oscillators

A mechanical or electrical system governed by a system of second order differential equations where the motion of each component influences the others. Coupled oscillators exhibit normal modes at characteristic frequencies determined by the eigenvalues of the stiffness matrix. They model multi-story building vibrations and multi-loop electrical circuits.

42. Coupled System of Differential Equations

A set of differential equations in which the derivatives of each dependent variable depend on the values of the other dependent variables. Coupled systems model interacting physical quantities such as currents in electrical networks, populations of competing species, and displacements in multi-degree-of-freedom mechanical systems.

43. Critical Damping

The condition in a second order linear system where the damping coefficient is exactly equal to twice the square root of the product of the mass and stiffness coefficients, resulting in the fastest return to equilibrium without oscillation. Critically damped systems represent the boundary between oscillatory and non-oscillatory behavior.

44. Critical Point

An equilibrium point of an autonomous system where all derivatives equal zero. The local behavior of solutions near a critical point is determined by linearization and the eigenvalues of the Jacobian matrix. Critical points are classified as nodes, spirals, saddles, or centers based on these eigenvalues.

45. d’Alembert’s Solution

The general solution to the one-dimensional wave equation on an infinite domain, expressed as the sum of a forward-traveling wave and a backward-traveling wave. Each wave is an arbitrary function of x minus ct and x plus ct respectively, and the specific functions are determined by the initial displacement and velocity profiles.

46. d-Operator Method

A technique for solving linear differential equations with constant coefficients using the differential operator D, where D raised to a power n represents the nth derivative. Algebraic operations on operators allow the particular solution to be written formally as the inverse operator applied to the nonhomogeneous term, often simplifying the computation.

47. Damped Natural Frequency

The frequency of oscillation in an underdamped second order system, equal to the natural frequency multiplied by the square root of one minus the square of the damping ratio. The damped natural frequency is always less than the natural frequency and decreases as damping increases until it reaches zero at critical damping.

48. Damped Oscillation

Oscillatory motion in which the amplitude decreases over time due to the presence of a resistive or dissipative force. Damped oscillations are described by solutions to second order linear equations with positive damping coefficients. The rate of decay depends on the damping ratio relative to the natural frequency.

49. Damping Ratio

A dimensionless parameter that characterizes the degree of damping in a second order system relative to critical damping. A damping ratio less than one indicates underdamping with oscillatory decay, equal to one indicates critical damping, and greater than one indicates overdamping with purely exponential decay.

50. Decay Constant

The positive constant in an exponential decay solution that determines how quickly the solution approaches zero. For a first order linear equation with a negative coefficient, the decay constant equals the absolute value of that coefficient divided by the leading coefficient. It is the reciprocal of the time constant.

51. Degree of a Differential Equation

The power to which the highest order derivative is raised after the equation has been cleared of radicals and fractions involving derivatives. The degree must be a positive integer. Most equations encountered in engineering are of degree one. The degree combined with the order determines the complexity of the equation.

52. Dependent Variable

The unknown function being solved for in a differential equation. It is typically denoted y and its derivatives appear in the equation. The dependent variable represents the physical quantity of interest such as displacement, temperature, current, or concentration.

53. Dependent Variable Transformation

The substitution of a new dependent variable to simplify or linearize a differential equation. Examples include the substitution that converts the Bernoulli equation to a linear one, the substitution that converts the Riccati equation near a known particular solution, and the amplitude-phase transformation for the harmonic oscillator.

54. Dirac Delta Function

A generalized function that is zero everywhere except at a single point where it is infinite, with a total integral of one. It models an impulsive force or instantaneous input in engineering systems. Its Laplace transform is simply one, and it appears in the solution of systems driven by sudden impulses.

55. Direction Field

A graphical representation of a first order differential equation created by drawing short line segments with slopes given by the equation at a grid of points in the plane. The direction field reveals the qualitative behavior of solutions without solving the equation and is used to sketch approximate solution curves.

56. Dirichlet Boundary Condition

A boundary condition that specifies the value of the solution itself at the boundary of the domain. For a heat conduction problem, a Dirichlet condition specifies the temperature at the boundary. It is the most common type of boundary condition and leads to well-posed problems for elliptic equations.

57. Dirichlet Problem

A boundary value problem for an elliptic partial differential equation such as Laplace’s equation, where the solution is specified on the entire boundary of the domain. The Dirichlet problem models steady-state temperature distribution with prescribed boundary temperatures and is one of the most important problems in mathematical physics.

58. Eigenfunction

A non-trivial solution to a differential equation boundary value problem that exists only for specific values of a parameter called the eigenvalue. Eigenfunctions form orthogonal sets that can be used to expand arbitrary functions in series solutions to partial differential equations. They are the spatial modes of vibration and heat conduction problems.

59. Eigenfunction Expansion

The representation of a function as an infinite series of eigenfunctions weighted by coefficients computed from the orthogonality relations. Eigenfunction expansions are used to solve boundary value problems for partial differential equations by separating the spatial and temporal parts of the solution.

60. Eigenvalue

A special value of a parameter in a differential equation boundary value problem for which a non-trivial solution exists. Eigenvalues determine the natural frequencies of vibrating systems, the decay rates of heat conduction modes, and the stability thresholds of equilibrium states. Computing eigenvalues is a central task in applied differential equations.

61. Eigenvalue Problem

A boundary value problem of the form involving a differential operator applied to a function equaling a parameter times the function with homogeneous boundary conditions. The problem asks for all values of the parameter (eigenvalues) and the corresponding non-trivial solutions (eigenfunctions). It is the continuous analog of the matrix eigenvalue problem.

62. Eigenvalue Stability Criterion

The principle that a linear system of differential equations is asymptotically stable if and only if all eigenvalues of the coefficient matrix have negative real parts. This criterion translates the stability question into an algebraic computation and is the standard tool for assessing stability in engineering control systems.

63. Equilibrium Stability Classification

The systematic categorization of equilibrium points based on the eigenvalues of the Jacobian matrix at the equilibrium. The classification identifies the equilibrium as a stable or unstable node, a stable or unstable spiral, a saddle point, or a center, each with a characteristic phase portrait pattern.

64. Existence Interval

The largest interval around the initial point on which the solution to an initial value problem is guaranteed to exist and be unique. For linear equations, the existence interval extends to the nearest point of discontinuity of the coefficients. For nonlinear equations, the solution may blow up before reaching a discontinuity.

65. Elimination Method

A technique for solving a system of differential equations by differentiating one or more equations and substituting to eliminate all dependent variables except one, reducing the system to a single higher-order equation in one unknown. The method mirrors algebraic elimination and is effective for small linear systems with constant coefficients.

66. Elliptic Partial Differential Equation

A class of second order partial differential equations characterized by having real, distinct characteristics that do not exist, meaning solutions are smooth throughout the domain. Laplace’s equation and Poisson’s equation are the standard examples. Elliptic equations model steady-state distributions such as equilibrium temperature and electrostatic potential.

67. Equilibrium Point

A constant solution to a differential equation or system where all derivatives are zero. The stability of an equilibrium point determines whether small perturbations grow or decay over time. Finding and classifying equilibrium points is the starting point for qualitative analysis of nonlinear systems.

68. Equilibrium Solution

A solution to a differential equation that is constant for all values of the independent variable. It satisfies the equation with all derivatives equal to zero. Equilibrium solutions represent steady states of physical systems and their stability determines the long-term behavior of nearby solutions.

69. Error Analysis in Numerical Methods

The study of how errors accumulate in numerical solutions to differential equations. Two types of error are distinguished: local truncation error, which arises from a single step, and global error, which accumulates over many steps. Understanding error behavior guides the selection of step size and numerical method for a given accuracy requirement.

70. Euler’s Method

The simplest numerical method for solving an initial value problem, which advances the solution by one step using the tangent line approximation at the current point. Each step computes the next value as the current value plus the step size times the derivative at the current point. Euler’s method is first-order accurate and introduces a local truncation error proportional to the square of the step size.

71. Exact Differential Equation

A first order differential equation of the form M times dx plus N times dy equals zero, where the expression is an exact differential of some function F. The test for exactness is that the partial derivative of M with respect to y equals the partial derivative of N with respect to x. The solution is F equals a constant.

72. Existence and Uniqueness Theorem

The theorem guaranteeing that an initial value problem has a unique solution in some interval around the initial point, provided the function and its partial derivative with respect to the dependent variable are continuous. This theorem is the theoretical foundation for the well-posedness of differential equation problems in engineering.

73. Explicit Solution

A solution to a differential equation in which the dependent variable is expressed directly as a function of the independent variable. Explicit solutions are the most convenient form for computation and analysis. Not all differential equations yield explicit solutions, and sometimes only implicit solutions can be obtained.

74. Exponential Order

A condition on a function requiring that it does not grow faster than an exponential function of the independent variable. Functions of exponential order have well-defined Laplace transforms. Most functions encountered in engineering applications, including polynomials and sinusoids, satisfy this condition.

75. Exponential Response Formula

A result giving the particular solution to a constant coefficient linear differential equation when the forcing function is an exponential, stating that the particular solution is the forcing exponential divided by the characteristic polynomial evaluated at the exponent. It provides the quickest route to the particular solution for exponential forcing.

76. Falling Body Problems

A class of first order differential equations modeling objects falling under gravity with air resistance proportional to velocity. The equation is separable and its solution describes the approach to terminal velocity. These problems are standard application examples in first order differential equations and appear in PRC engineering board exams.

77. Final Value

The limiting value of the solution as the independent variable approaches infinity, provided the limit exists. The final value theorem of Laplace transforms computes this limit directly from the transform without inverting it. The final value represents the steady-state output of an engineering system subjected to a persistent input.

78. Final Value Theorem

A result in Laplace transform theory stating that the limit of a function as time approaches infinity equals the limit of s times its Laplace transform as s approaches zero, provided the limit exists. It allows the steady-state value of a system response to be determined directly from the transform without computing the inverse.

79. First Order Differential Equation

A differential equation involving only the first derivative of the unknown function. First order equations are the most basic class and include linear, separable, exact, Bernoulli, and Riccati types. They model a wide range of physical phenomena including radioactive decay, population growth, cooling, and simple electrical circuits.

80. First Order Linear Differential Equation

A linear differential equation of the form where the derivative of y plus a function of x times y equals another function of x. It is solved using an integrating factor equal to e raised to the integral of the coefficient function. This is one of the most important and frequently tested equation types in engineering mathematics.

81. First Order System

A system of differential equations in which all equations involve only first derivatives. Any higher order differential equation can be converted to an equivalent first order system by introducing auxiliary variables for each derivative. First order systems are the standard form for numerical solution and for matrix analysis of linear systems.

82. Fixed Point

Another term for equilibrium point in the study of autonomous systems. A fixed point is a state that does not change over time. The classification and stability of fixed points through linearization is a foundational technique in engineering control and dynamical systems analysis.

83. Forced Oscillation

Oscillatory motion driven by an external periodic force. The governing equation is a second order nonhomogeneous linear equation. The complete solution consists of the transient complementary function that decays over time and the steady-state particular solution that persists at the forcing frequency.

84. Fourier Series Solution

The representation of the solution to a partial differential equation as a series of sinusoidal functions, obtained by applying separation of variables and expanding the initial or boundary data in a Fourier series. This method is the standard approach for solving the heat equation and wave equation on finite domains.

85. Fourier Transform

An integral transform that decomposes a function defined on the entire real line into its frequency components. It converts differential equations with constant coefficients into algebraic equations in the frequency domain. The Fourier transform is particularly useful for solving partial differential equations on infinite or semi-infinite domains.

86. Free Oscillation

Oscillatory motion of a system in the absence of an external driving force, governed by the homogeneous differential equation. The frequency of free oscillation is determined by the physical parameters of the system. Free oscillations may be undamped, underdamped, critically damped, or overdamped depending on the system properties.

87. Frobenius Method

A technique for finding power series solutions to second order linear differential equations with regular singular points. The method assumes a solution of the form x raised to a power r times a power series and substitutes it into the equation to determine r and the coefficients. The two values of r are roots of the indicial equation.

88. Fundamental Matrix

A matrix whose columns are linearly independent solutions to a system of first order linear differential equations. The fundamental matrix is used to express the general solution of the system and to construct the variation of parameters formula for nonhomogeneous systems. It plays the role of the Wronskian for systems.

89. Fundamental Set of Solutions

A set of linearly independent solutions to a homogeneous linear differential equation that spans the solution space. For an nth order equation, the fundamental set contains exactly n solutions. Every solution to the homogeneous equation can be written as a linear combination of the fundamental set.

90. Fundamental Theorem of ODEs

The existence and uniqueness theorem for initial value problems, which establishes that under appropriate continuity conditions a unique solution exists in a neighborhood of the initial point. This theorem is the cornerstone of the theory of ordinary differential equations and justifies all practical solution methods.

91. General Solution

The complete family of solutions to a differential equation, expressed with arbitrary constants whose number equals the order of the equation. The general solution to a nonhomogeneous equation is the sum of the complementary function and a particular solution. Every specific solution is obtained by assigning values to the arbitrary constants.

92. Global Existence

The property of a differential equation solution that exists and remains bounded for all values of the independent variable, not just in a neighborhood of the initial point. Global existence is guaranteed for linear equations on intervals where the coefficients are continuous but must be established separately for nonlinear equations.

93. Gompertz Equation

A nonlinear first order differential equation used to model bounded growth, particularly in the modeling of tumor growth and population dynamics. The equation incorporates a growth rate that decreases logarithmically as the population approaches a carrying capacity, producing a characteristic S-shaped curve with an asymmetric inflection point.

94. Green’s Function

A function that represents the response of a linear differential equation to a unit impulse input. The solution to the equation for an arbitrary forcing function is obtained by convolving the Green’s function with the forcing function. Green’s functions are fundamental in the theory of linear differential equations and appear in advanced engineering analysis.

95. Growth and Decay Models

First order differential equations modeling exponential growth or decay, where the rate of change is proportional to the current quantity. The solution is an exponential function. Standard engineering applications include radioactive decay, population growth, compound interest, and cooling or heating governed by Newton’s law.

96. Harmonic Oscillator

A second order linear differential equation with constant coefficients modeling a mass-spring system or an LC circuit. In its undamped form, the equation describes simple harmonic motion with a constant amplitude. The natural frequency is determined by the ratio of the stiffness to the mass. It is one of the most important models in engineering dynamics.

97. Heat Equation

A second order linear partial differential equation describing how temperature evolves over time in a conducting medium. It is a parabolic equation solved using separation of variables and Fourier series for finite domains. The heat equation models diffusion processes in heat transfer, mass transfer, and financial engineering.

98. Heaviside Step Function

A piecewise function equal to zero for negative arguments and one for positive arguments, used to represent sudden switches in engineering inputs. Its Laplace transform is one divided by s. The Heaviside function and its shifted versions are used to build piecewise forcing functions in Laplace transform solutions.

99. Homogeneous Differential Equation

A differential equation in which every term contains the dependent variable or its derivatives, with no term depending solely on the independent variable. The solution space of a homogeneous linear equation is a vector space. Setting the right-hand side to zero always yields the associated homogeneous equation.

100. Homogeneous Function

A function where scaling all arguments by a constant factor scales the function value by a power of that constant. First order equations involving homogeneous functions of x and y can be reduced to separable equations by the substitution v equals y divided by x. This technique is a standard method for a specific class of first order equations.

251. Zeros of the Characteristic Polynomial

The roots of the characteristic equation, which determine the natural modes of a linear differential equation system. Real negative roots give decaying exponentials, positive roots give growing exponentials, and complex roots give oscillatory exponentials. The location of the zeros in the complex plane determines the stability and character of the system response.

Conclusion

Differential equations is one of the most interconnected subjects in engineering mathematics. The concepts build on each other in a logical chain: understanding linearity leads to superposition, superposition leads to the complementary function, and the complementary function combined with a particular solution gives the complete answer. If you invest the time to understand why each method works, not just how to apply it, you will find that the subject becomes much more manageable under exam pressure. The Laplace transform in particular is worth mastering deeply because it handles all the standard equation types, automatically incorporates initial conditions, and deals elegantly with discontinuous forcing functions that would be tedious to handle by other methods.

For the PRC board exam, prioritize the following in your review. First order equations: linear, separable, and exact appear frequently as standalone problems and as embedded steps in larger application problems. Second order constant coefficient equations with their characteristic root analysis and the method of undetermined coefficients account for a large share of the vibration and circuit problems. The Laplace transform section typically produces several questions involving step functions, impulse inputs, and initial value problems with discontinuous forcing. Systems of differential equations appear in more advanced problems involving multiple loops in circuits or coupled mechanical systems. And numerical methods, particularly Euler and Runge-Kutta, show up when problems provide tabular data or ask for approximations.

Differential equations rewards systematic preparation more than any other subject in engineering mathematics. The number of techniques is large but finite, and each technique applies to a clearly defined class of equations. Knowing the classification of an equation its order, linearity, coefficient type, and homogeneity immediately narrows the applicable methods. Build this classification habit first, then practice each method until the mechanical steps are automatic. Application problems become straightforward once you can translate physical descriptions into differential equations confidently. The 251 terms in this list give you the vocabulary and conceptual framework to build that confidence efficiently throughout your review cycle.

For practice problems on all these topics, head over to our Differential Equations Problems and Solutions section here on PinoyBix. Hundreds of solved exam-type questions, complete with step-by-step solutions, organized by topic so you can drill exactly what you need to work on.