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

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