Ordinary Differential Equation - Summary of Exact Solutions

Summary of Exact Solutions

Some differential equations have solutions which can be written in an exact and closed form. Several important classes are given here.

In the table below, P(x), Q(x), P(y), Q(y), and M(x,y), N(x,y) are any integrable functions of x, y, and b and c are real given constants, and C₁, C₂,... are arbitrary constants (complex in general). The differential equations are in their equivalent and alternative forms which lead to the solution through integration.

In the integral solutions, λ and ε are dummy variables of integration (the continuum analogues of indices in summation), and the notation ∫xF(λ)dλ just means to integrate F(λ) with respect to λ, then after the integration substitute λ = x, without adding constants (explicitly stated).

Differential equation	Solution method	General solution
Separable equations
First order, separable in x and y (general case, see below for special cases)	Separation of variables (divide by P2Q1).
First order, separable in x	Direct integration.
First order, autonomous, separable in y	Separation of variables (divide by F).
First order, separable in x and y	Integrate throughout.
General first order equations
First order, homogeneous	Set y = ux, then solve by separation of variables in u and x.
First order, separable	Separation of variables (divide by xy).	If N = M, the solution is xy = C.
Exact differential, first order where	Integrate throughout.	where Y(y) and X(x) are functions from the integrals rather than constant values, which are set to make the final function F(x, y) satisfy the initial equation.
Inexact differential, first order where	Integration factor μ(x, y) satisfying	If μ(x, y) can be found:
General second order equations
Second order	Multiply equation by 2dy/dx, substitute, then integrate with respect to x, then y.
Second order, autonomous	Multiply equation by 2dy/dx, substitute, then integrate twice.
Linear equations (up to nth order)
First order, linear, inhomogeneous, function coefficients	Integrating factor: .
Second order, linear, inhomogeneous, constant coefficients	Complementary function yc: assume yc = eαx, substitute and solve polynomial in α, to find the linearly independent functions . Particular integral yp: in general the method of variation of parameters, though for very simple r(x) inspection may work.	If b2 > 4c, then: If b2 = 4c, then: If b2 < 4c, then:
nth order, linear, inhomogeneous, constant coefficients	Complementary function yc: assume yc = eαx, substitute and solve polynomial in α, to find the linearly independent functions . Particular integral yp: in general the method of variation of parameters, though for very simple r(x) inspection may work.	Since αj are the solutions of the polynomial of degree n:, then: for αj all different, for each root αj repeated kj times, for some αj complex, then setting α = χj + iγj, and using Euler's formula, allows some terms in the previous results to be written in the form where ϕj is an arbitrary constant (phase shift).