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A first-order ordinary differential equation relates an unknown function to its first derivative. For an equation of the form

if can be expressed using separation of variables as

then the equation can be expressed as

where . Integrating both sides gives

Each zero of also gives the constant function solution .

For a first-order linear ordinary differential equation

assume and are continuous functions on an interval containing . Multiplication by an integrating factor satisfying turns the left-hand side into by the product rule. The choice

is nonzero and satisfies . Integrating gives the solution with initial conditions specified by ,

Varying gives the general solution.

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.

See also

Exact First-Order Ordinary Differential Equation, Integrating Factor, Ordinary Differential Equation, Second-Order Ordinary Differential Equation, Separation of Variables, Variation of Parameters

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References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 440-445, 1985.

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First-Order Ordinary Differential Equation

Cite this as:

Weisstein, Eric W. "First-Order Ordinary Differential Equation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/First-OrderOrdinaryDifferentialEquation.html

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