Question
Solve the following differential equations:$\text{ye}^{\frac{\text{x}}{\text{y}}}\text{dx}=(\text{xe}^{\frac{\text{x}}{\text{y}}}+\text{y}^2)\text{dy, y}\neq0$
✓
Answer
$\text{ye}^{\frac{\text{x}}{\text{y}}}\text{dx}=(\text{xe}^{\frac{\text{x}}{\text{y}}}+\text{y}^2)\text{dy}$
$\Rightarrow\text{ye}^{\frac{\text{x}}{\text{y}}}\text{dx = xe}^{\frac{\text{x}}{\text{y}}}\text{dy}+\text{y}^2\text{dy}$
$\Rightarrow\text{ye}^{\frac{\text{x}}{\text{y}}}\text{dx}-\text{xe}^{\frac{\text{x}}{\text{y}}}\text{dy = y}^2\text{dy}$
$\Rightarrow(\text{ydx}-\text{xdy})\text{e}^{\frac{\text{x}}{\text{y}}}=\text{y}^2\text{dy}$
$\Rightarrow\frac{(\text{ydx}-\text{xdy})}{\text{y}^2}\text{e}^{\frac{\text{x}}{\text{y}}}=\text{dy}$
$\Rightarrow\text{e}^{\frac{\text{x}}{\text{y}}}\text{d}\Big(\frac{\text{x}}{\text{y}}\Big)=\text{dy}$
$\Rightarrow\int\text{e}^{\frac{\text{x}}{\text{y}}}\text{d}\Big(\frac{\text{x}}{\text{y}}\Big)=\int\text{dy}$
$\Rightarrow\text{e}^{\frac{\text{x}}{\text{y}}}=\text{y + C}$
$\Rightarrow\text{ye}^{\frac{\text{x}}{\text{y}}}\text{dx = xe}^{\frac{\text{x}}{\text{y}}}\text{dy}+\text{y}^2\text{dy}$
$\Rightarrow\text{ye}^{\frac{\text{x}}{\text{y}}}\text{dx}-\text{xe}^{\frac{\text{x}}{\text{y}}}\text{dy = y}^2\text{dy}$
$\Rightarrow(\text{ydx}-\text{xdy})\text{e}^{\frac{\text{x}}{\text{y}}}=\text{y}^2\text{dy}$
$\Rightarrow\frac{(\text{ydx}-\text{xdy})}{\text{y}^2}\text{e}^{\frac{\text{x}}{\text{y}}}=\text{dy}$
$\Rightarrow\text{e}^{\frac{\text{x}}{\text{y}}}\text{d}\Big(\frac{\text{x}}{\text{y}}\Big)=\text{dy}$
$\Rightarrow\int\text{e}^{\frac{\text{x}}{\text{y}}}\text{d}\Big(\frac{\text{x}}{\text{y}}\Big)=\int\text{dy}$
$\Rightarrow\text{e}^{\frac{\text{x}}{\text{y}}}=\text{y + C}$
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