Solve the differential equation : (d y d x )=2 x-3 y - Vidyadip

Question

Solve the differential equation : $\log \left(\frac{d y}{d x}\right)=2 x-3 y$

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Answer

Given, $\log \left(\frac{d y}{d x}\right)=2 x-3 y$
$\Rightarrow \quad \frac{d y}{d x}=e^{2 x-3 y}$
$\Rightarrow \quad \frac{d y}{d x}=\frac{e^{2 x}}{e^{3 y}}$
$\Rightarrow \quad e^{3 y} d y=e^{2 x} d x$
Integrating both sides, we get
$\int e^{3 y} d y=\int e^{2 x} d x$
$\Rightarrow \frac{e^{3 y}}{3}=\frac{e^{2 x}}{2}+C$
is the required solution.

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