# Solve the following first-order linear differential equations? y'−4x^3y=8x^3

Solve the following first-order linear differential equations? $y\prime -4{x}^{3}y=8{x}^{3}$
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The ODE is
$y\prime -4{x}^{3}y=8{x}^{3}$
$\frac{dy}{dx}=8{x}^{3}+4{x}^{3}y=4{x}^{3}\left(2+y\right)$
$\frac{dy}{y+2}=4{x}^{3}dx$
$\int \frac{dy}{y+2}=\int 4{x}^{3}dx$
$\mathrm{ln}\left(y+2\right)={x}^{4}+{C}_{1}$
$y+2={e}^{{x}^{4}+{C}_{1}}={e}^{{C}_{1}}{e}^{{x}^{4}}=C{e}^{{x}^{4}}$
$y=C{e}^{{x}^{4}}-2$