Question

The solution of $y=x\left(\frac{dy}{dx}\right)+{\left(\frac{dy}{dx}\right)}^3$ are given by (where $p=dy/dx$ and $k$ is constant).

• A. $y=0$
• B. $y=kx+k^3$
• C. $y=k{e}^{x+k}$
• D. None of these

Answer 1

The correct option is A $y=0$

Put $p=\frac{dy}{dx}$ in the given equation $y=x\left(\frac{dy}{dx}\right)+{\left(\frac{dy}{dx}\right)}^3$.

$y=px+p^3$

$\frac{dy}{dx}=\frac{dp}{dx}\left(x\right)+p+3p^2\frac{dp}{dx}$

$p=x\frac{dp}{dx}+p+3p^2\frac{dp}{dx}$

$\frac{dp}{dx}\left(x+3p^2\right)=0$

$\frac{dp}{dx}=0;x+3p^2=0$

$p=0;p^2=-\frac{x}{3}$

$\frac{dy}{dx}=0$

$y=0$