Solve the following differential equation d y d x = x ^2 y + y - Vidyadip

Question

Solve the following differential equation $\frac{ d y}{ d x}= x ^2 y + y$

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Answer

$ \frac{ d y}{ d x}= x ^2 y + y$
$\therefore \frac{ d y}{ d x}= y \left( x ^2+1\right)$
$\therefore \frac{ d y}{y}=\left( x ^2+1\right) dx $
Integrating on both sides, we get
$ \int \frac{ d y}{y}=\int\left(x^2+1\right) d x$
$\therefore \log | y |=\frac{x^3}{3}+x+ c $

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