Question

A solution of $y=2x\left(\frac{dy}{dx}\right)+x^2(\frac{dy}{dx}{)}^4$ is

• A. $y=2c^{\frac{1}{2}}{x}^{\frac{1}{4}}+c$
• B. $y=2\sqrt{c{x}^2}+c^2$
• C. $y=2\sqrt{c}(x+1)$
• D. $y=2\sqrt{cx}+c^2$

Answer 1

The correct option is D $y=2\sqrt{cx}+c^2$

Write $p=\frac{dy}{dx}$ and differentiating w.r.t.x, we have
$p=2p+2x\frac{dp}{dx}+2x{p}^4+4p^4+4p^3{x}^2\frac{dp}{dx}\Rightarrow 0=p(1+2x{p}^3)+2x(1+2p^{3}x)\Rightarrow p+2x\frac{dp}{dx}=0\Rightarrow 2\frac{dp}{dx}=-\frac{dx}{x}\Rightarrow 2logp+logx=const\Rightarrow p^{2}x=corp=\sqrt{\frac{c}{x}}$
Substituting this value in the given equation, we get $y=2\sqrt{cx}+c^2$