Examples of Differential Equations - Non-separable (non-homogeneous) First-order Linear Ordinary Differential Equations

Non-separable (non-homogeneous) First-order Linear Ordinary Differential Equations

First-order linear non-homogeneous ODEs (ordinary differential equations) are not separable. They can be solved by the following approach, known as an integrating factor method. Consider first-order linear ODEs of the general form:

The method for solving this equation relies on a special integrating factor, μ:

We choose this integrating factor because it has the special property that its derivative is itself times the function we are integrating, that is:

Multiply both sides of the original differential equation by μ to get:

Because of the special μ we picked, we may substitute dμ/dx for μ p(x), simplifying the equation to:

Using the product rule in reverse, we get:

Integrating both sides:

Finally, to solve for y we divide both sides by :

Since μ is a function of x, we cannot simplify any further directly.