# Integrating factor

From Example Problems

In the case of a linear first order homogeneous ODE such as the equation should always be divided by so the coefficient of the term is unity.

The integrating factor () is found by exponentiating the integral of the coefficient of the term with respect to . The integral is indefinate and you don't have to worry about constants of integration.

Now, multiply the equation by the integrating factor.

Notice that the first two terms of the left hand side (LHS) of the equation are the chain rule expansion of

Integrate both sides with respect to x, and here constants of integration are required again.

Plugging in 0 for the IC,

So the unique solution for this IC is