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Q.
Let $y=f(x)$ satisfies the differential equation $x y(1+y) d x=d y$. If $f(0)=1$ and $f(2)=\frac{e^2}{k-e^2}$ then $k$ is equal to
Differential Equations
Solution:
$\frac{d y}{d x}=x y(1+y) \Rightarrow \int \frac{d y}{y(1+y)}=\int x d x \Rightarrow \frac{2 y}{1+y}=e^{\frac{x^2}{2}} [$ As $f(0)=1]$
$\therefore f (2)=\frac{ e ^2}{2- e ^2}$