Question

Separable D.E

Answer 1

\[x^3(y)\prime+2y=e^\frac{ 1 }{ x^2 }\]

Answer 2

if y(1)=e

Answer 3

@TuringTest

Answer 4

@iambatman

Answer 5

This doesn't look separable to me... But it is linear.
\[\large\begin{align*}
x^3y'+2y&=e^{1/x^2}\\\\
y'+2x^{-3}y&=x^{-3}e^{x^{-2}}
\end{align*}\]
Integrating factor:
\[\begin{align*}\log\mu(x)&=\int2x^{-3}~dx\\\\
&=-x^{-2}\\\\
\mu(x)&=e^{-x^{-2}}
\end{align*}\]
Distribute the IF:
\[\large\begin{align*}
e^{-x^{-2}}y'+2x^{-3}e^{-x^{-2}}y&=x^{-3}\\\\
\frac{d}{dx}\left[e^{-x^{-2}}y\right]&=x^{-3}\\\\
e^{-x^{-2}}y&=\int x^{-3}~dx\\\\
e^{-x^{-2}}y&=-\frac{1}{2}x^{-2}+C\\\\
y&=-\frac{1}{2}x^{-2}e^{x^{-2}}+Ce^{x^{-2}}
\end{align*}\]