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.
Read more about this topic: Examples Of Differential Equations
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