Question

The solutions of $y=x(\frac{dy}{dx}+{\left(\frac{dy}{dx}\right)}^3)$ are given by

• A. The constant function $y=0$.
• B. $y=p^{-3}{e}^{\frac{1}{2p^2}}(p+p^3)$
• C. $y=p^3{e}^{-\frac{1}{2}{p}^2}(p+p^3)$
• D. $y{e}^{-\frac{1}{2}{p}^2}=p^{-2}+1$

Answer 1

Answer: (A), (B)

The constant function $y=0$.
$y=p^{-3}{e}^{\frac{1}{2p^2}}(p+p^3)$

$Clearlytheconstantfunctiony=0isasolution.Differentiatingthegivenequationw.r.t.x,letp=\frac{dy}{dx}p=(p+p^3)+x(\frac{dp}{dx}+3p^2\frac{dp}{dx})-p^3=x(1+3p^2)\frac{dp}{dx}-\frac{dx}{x}=(\frac{1}{p^3}+\frac{3}{p})dp\log p^{3}x=\frac{1}{2p^2}+Cx{p}^3=K{e}^{\frac{1}{2p^2}}Puttingvalueofxisinthegivenequation,wehavey=K{p}^{-3}{e}^{\frac{1}{2p^2}}(p+p^3)$