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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

$\frac{dy}{dx} + f(x)y = g(x)$,

the solution can be found with the formula

$y = \frac{1}{\mu} \int \mu g(x) dx, \mu = e^{\int f(x)dx}$

For example:

$\frac{dy}{dx} = 2x^2 + x^2y$
$\frac{dy}{dx} - x^2y= 2x^2$
$\frac{1}{\mu} \int (\mu) (2x^2) dx, \mu = e^{-\frac{1}{3} x^3}$
$2e^{\frac{1}{3} x^3} \int e^{-\frac{1}{3} x^3} x^2 dx$

By using integration by substitution, we get:

$y = 2e^{\frac{1}{3} x^3} ( -e^{-\frac{1}{3} x^3} + C ) = Ce^{\frac{1}{3} x^3} - 2$