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Q.
If $\frac{dy}{dx}+\frac{y}{x}=x^{2}$, then $2y\left(2\right)-y\left(1\right)=$
Differential Equations
Solution:
It is linear differential equation with
$I.F.= \text{exp} \int \frac{dx}{x}=x$
$\therefore $ Solution is, $xy=\int x^{2}\,xdx=\frac{x^{4}}{4}+c$
Now, $x=2$
$\Rightarrow 2y\left(2\right)=\frac{16}{4}+c$
and $x=1$
$\Rightarrow y\left(1\right)=\frac{1}{4}+c$
$\Rightarrow 2y\left(2\right)-y\left(1\right)=\frac{15}{4}$.