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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. Community content is available under CC-BY-SA unless otherwise noted.