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  4. Let y=f(x) be the solution of the differential equation y(x+1) d x-x2 d y=0, y(1)=e. Then displaystyle lim x arrow 0+ f(x) is equal to

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Q. Let $y=f(x)$ be the solution of the differential equation $y(x+1) d x-x^2 d y=0, y(1)=e$. Then $\displaystyle\lim _{x \rightarrow 0^{+}} f(x)$ is equal to

JEE MainJEE Main 2023Differential Equations

Solution:

$ \frac{x+1}{x^2} d x=\frac{d y}{y} $
$ \ln x-\frac{1}{x}=\ln y+c$
$ (1, e ) $
$ c =-2 $
$ \ln x-\frac{1}{x}=\ln y-2 $
$ y=e^{\ln x}-\frac{1}{x}+2 $
$ \displaystyle\lim _{x \rightarrow 0^{+}} e^{\ln x-1}-\frac{1}{x}+2$
$ =e^{-\infty} $
$ =0$