Question

If $y\frac{dy}{dx}=xâ\frac{y^2}{x^2}+\frac{Ï•\left(\frac{y^2}{x^2}\right)}{{Ï•}^{′}\left(\frac{y^2}{x^2}\right)}â,x>0,Ï•>0$, and $y(1)=−1$ then $Ï•\left(\frac{y^2}{4}\right)$ is equal to :

• A. $4Ï•(2)$
• B. $4Ï•(1)$
• C. $2Ï•(1)$
• D. $Ï•(1)$

Answer 1

Answer: (B)

Let, $y=tx$
$\frac{dy}{dx}=t+x\frac{dt}{dx}$
$∴tx\left(t+x\frac{dt}{dx}\right)=x\left(t^2+\frac{φ\left(t^2\right)}{{φ}^{′}\left(t^2\right)}\right)$
$t^2+xt\frac{dt}{dx}=t^2+\frac{φ\left(t^2\right)}{{φ}^{′}\left(t^2\right)}$
$∫\frac{t{φ}^{′}\left(t^2\right)}{φ\left(t^2\right)}dt=∫\frac{dx}{x}$
Let $φ\left(t^2\right)=p$
$∴{φ}^{′}\left(t^2\right)2tdt=dp$
$⇒∫\frac{dy}{2p}=∫\frac{dx}{x}$
$\frac{1}{2}ℓnφ\left(t^2\right)=ℓnx+ℓnc$
$φ\left(t^2\right)=x^{2}k$
$φ\left(\frac{y^2}{x^2}\right)=k{x}^2,φ(1)=k$
$φ\left(\frac{y^2}{4}\right)=4φ(1)$