A linear differential equation is a differential equation (either ordinary or partial) where each function and derivative of y (or any dependent variable) has an exponent of either one or zero.

For linear equations in the form

,

the solution can be found with the formula

For example:

By using integration by substitution, we get:


Proof of formula

Suppose we have a linear differential equation in the form

Let's multiply this by an integrating factor

Let's assume that the left hand side is now integratable using the reverse product rule, and then find the conditions for that make this possible.

The product rule says for two differentiable functions,

In our case, , and , and this means we require

Solving this separable differential equation is simple and yields

Going back to our earlier equation, we can now proceed and reverse the product rule

Integrating and dividing by gives us the formula above.


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