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

Question

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

✓

Answer

$
\begin{aligned}
& \frac{d y}{d x}= x ^2 y + y \\
& \therefore \frac{d y}{d x}= y \left( x ^2+1\right) \\
& \therefore \frac{1}{y} dy =\left( x ^2+1\right) dx
\end{aligned}
$
Integrating, we get
$
\begin{aligned}
& \int \frac{1}{y} d y=\int\left(x^2+1\right) d x \\
& \therefore \log |y|=\frac{x^3}{3}+x+c
\end{aligned}
$
This is the general solution.

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