Non-homogeneous ODE

Non-homogeneous ODEs are where the differential equation looks like:

$$a_n(t)x^n+a_{n-1}(t)x^{n-1}+\cdots +a_1(t)x^{\prime }+a_0(t)x=g(t)$$

where $g(t)\ne 0$. If it was 0, we would be dealing with a homogeneous ODE. We often call $g(t)$ some forcing function.

The general solution of any non-homogeneous linear ODE is given by the sum of the general solution $x_c(t)$ of the associated homogeneous equation and the particular solution $x_p(t)$ of the non-homogeneous equation.

$$x(t)=x_c(t)+x_p(t)$$

There are no systematic universal methods for finding the particular solution to non-homogeneous ODEs, even when there are linear terms. In general, the form of $x_p(t)$ should be motivated by the terms of $g(t)$. Strategies to find the particular solution include:

• Method of undetermined coefficients
• Variation of parameters