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  4. If y (d y/d x)=x[(y2/x2)+(φ((y2/x2))/φ prime((y2)x2))], x > 0, φ > 0, and y(1)=-1 then φ(( y 2/4)) is equal to :

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Q. If $y \frac{d y}{d x}=x\left[\frac{y^{2}}{x^{2}}+\frac{\phi\left(\frac{y^{2}}{x^{2}}\right)}{\phi^{\prime}\left(\frac{y^{2}}{x^{2}}\right)}\right], x > 0, \phi > 0$, and $y(1)=-1$ then $\phi\left(\frac{ y ^{2}}{4}\right)$ is equal to :

JEE MainJEE Main 2021Differential Equations

Solution:

Let, $y=t x$
$\frac{d y}{d x}=t+x \frac{d t}{d x}$
$\therefore t x\left(t+x \frac{d t}{d x}\right)=x\left(t^{2}+\frac{\varphi\left(t^{2}\right)}{\varphi^{\prime}\left(t^{2}\right)}\right)$
$t ^{2}+ xt \frac{ dt }{ dx }= t ^{2}+\frac{\varphi\left( t ^{2}\right)}{\varphi^{\prime}\left( t ^{2}\right)}$
$\int \frac{ t \varphi^{\prime}\left( t ^{2}\right)}{\varphi\left( t ^{2}\right)} dt =\int \frac{ dx }{ x }$
Let $\varphi\left( t ^{2}\right)= p$
$\therefore \varphi^{\prime}\left( t ^{2}\right) 2 tdt = dp$
$\Rightarrow \int \frac{ dy }{2 p }=\int \frac{ dx }{ x }$
$\frac{1}{2} \ell n \varphi\left( t ^{2}\right)=\ell n x +\ell nc$
$\varphi\left( t ^{2}\right)= x ^{2} k$
$\varphi\left(\frac{ y ^{2}}{ x ^{2}}\right)= kx ^{2}, \varphi(1)= k$
$\varphi\left(\frac{ y ^{2}}{4}\right)=4 \varphi(1)$