First-Order Ordinary Differential Equation

Given a first-order ordinary differential equation

if can be expressed using separation of variables as

then the equation can be expressed as

and the equation can be solved by integrating both sides to obtain

Any first-order ODE of the form

can be solved by finding an integrating factor such that

Dividing through by yields

However, this condition enables us to explicitly determine the appropriate for arbitrary and . To accomplish this, take

in the above equation, from which we recover the original equation (◇), as required, in the form

But we can integrate both sides of (9) to obtain

Now integrating both sides of (◇) gives

(with now a known function), which can be solved for to obtain

where is an arbitrary constant of integration.

Given an th-order linear ODE with constant coefficients

first solve the characteristic equation obtained by writing

and setting to obtain the complex roots.

Factoring gives the roots ,

For a nonrepeated real root , the corresponding solution is

If a real root is repeated times, the solutions are degenerate and the linearly independent solutions are

Complex roots always come in complex conjugate pairs, . For nonrepeated complex roots, the solutions are

If the complex roots are repeated times, the linearly independent solutions are

Linearly combining solutions of the appropriate types with arbitrary multiplicative constants then gives the complete solution. If initial conditions are specified, the constants can be explicitly determined. For example, consider the sixth-order linear ODE

which has the characteristic equation

The roots are 1, 2 (three times), and , so the solution is

If the original equation is nonhomogeneous (), now find the particular solution by the method of variation of parameters. The general solution is then

where the solutions to the linear equations are , , ..., , and is the particular solution.