Solve the differential equation: (x+1) d y-2 x y d x=0 - Vidyadip

Question

Solve the differential equation: $(x+1) d y-2 x y d x=0$

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Answer

$(x+1) d y-2 x y d x=0$
$\Rightarrow \quad \int \frac{d y}{y}=\int \frac{2 x}{x+1} d x$
$\Rightarrow \quad \log y=2 \int \frac{x+1-1}{x+1} d x$
$=2 \int\left(1-\frac{1}{x+1}\right) d x$
$\Rightarrow \log y=2[x-\log (x+1)]+\log c$
$\log y+2 \log (x+1)-\log c=2 x$
$\Rightarrow \quad \log \frac{y(x+1)^2}{c}=2 x$
$\Rightarrow \quad \frac{y(x+1)^2}{c}=e^{2 x}$
$\Rightarrow \quad y=\frac{c e^{2 x}}{(x+1)^2}$

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