 # What is the general solution of the differential equation? : dy/dx+3y=3x^2e^(−3x) Liam Keller 2022-09-13 Answered
What is the general solution of the differential equation? : $\frac{dy}{dx}+3y=3{x}^{2}{e}^{-3x}$
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$\frac{dy}{dx}+3y=3{x}^{2}\cdot {e}^{-3x}$
$\frac{dy}{dx}\cdot {e}^{3x}+3y\cdot {e}^{3x}=3{x}^{2}\cdot {e}^{-3x}\cdot {e}^{3x}$
$\frac{d}{dx}\left(y\cdot {e}^{3x}\right)=3{x}^{2}$
$y\cdot {e}^{3x}={x}^{3}+C$
$y=\left({x}^{3}+C\right)\cdot {e}^{-3x}$
Note: This differential equation is first order and linear one.
1) I multiplied both sides with ${e}^{3x}$ for converting left side to the exact differential equation.
2) I integrated both sides.
3) I multiplied with ${e}^{-3x}$ for solving y.

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