Analytic Methods for Solving First-Order ODE

The general form of a first-order differential equation is

Here t is the independent variable and y is the dependent variable. The goal is to find the unknown function y(t).

Classes of First-Order ODE

It is said that a differential equation is solved exactly if the answer can be expressed in the form of an integral. The following classes of first-order ode can be solved exactly:

• directly integrable

  

  (derivative function depends on independent variable only)

• linear

  

• separable

  

• homogeneous

  

• exact

  

  where the partial derivatives satisfy f_y=g_t.

• Bernoulli

  

  where n doesn't equal 1.

This list is not exhaustive, but covers core classes of first-order ode that can
be solved exactly. If an ode cannot be solved by analytic methods, then it can
always be approximated by a numerical method.

[ODE Home] [1st-Order Home] [2nd-Order Home] [Laplace Transform Home] [Notation] [References]

Copyright © 1996 Department of Mathematics, Oregon State University

If you have questions or comments, don't hestitate to contact us.