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Q.
If $y(x)$ is solution of $(x+1) \frac{d y}{d x}-x y=1, y(0)=-1$, then $y\left(\frac{-6}{5}\right)$ is equal to
Differential Equations
Solution:
We have $\frac{ dy }{ dx }-\frac{ x }{ x +1} \cdot y =\frac{1}{ x +1} $ (linear differential equation) So, general solution is
$y(x+1) e^{-x}=\int e^{-x} d x+C \Rightarrow y(x+1) e^{-x}=-e^{-x}+C$
As $y (0)=-1$, gives $C =0$
$\Rightarrow y ( x +1)=-1 \Rightarrow y =\frac{-1}{1+ x } $
$\therefore y \left(\frac{-6}{5}\right)=5$