Question

solve dy/dx=e^3x-2y+ x^2 e^-2y

Answer 1

$\textbf{Given:}$

$\text{We apply variable separable method to solve the}$

$\text{given differential equation}$

$\displaystyle\frac{dy}{dx}=e^{3x-2y}+x^2e^{-2y}$

$\displaystyle\frac{dy}{dx}=e^{3x}^{-2y}+x^2e^{-2y}$

$\displaystyle\frac{dy}{dx}=(e^{3x}+x^2)e^{-2y}$

$\implies\displaystyle\frac{1}{e^{-2y}}\frac{dy}{dx}=(e^{3x}+x^2)$

$\implies\displaystyle\,e^{2y}\frac{dy}{dx}=(e^{3x}+x^2)$

$\implies\displaystyle\,e^{2y}\,dy=(e^{3x}+x^2)dx$

$\text{Integrating}$

$\displaystyle\int\,e^{2y}\,dy=\int(e^{3x}+x^2)dx$

$\displaystyle\bf\,\frac{e^{2y}}{2}=\frac{e^{3x}}{3}+\frac{x^3}{3}+c$

$\therefore\textbf{The solution is \bf\,\frac{e^{2y}}{2}=\frac{e^{3x}}{3}+\frac{x^3}{3}+c }$

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