24.27 problem 689

Internal problem ID [3428]

Book: Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section: Various 24
Problem number: 689.
ODE order: 1.
ODE degree: 1.

CAS Maple gives this as type [[_homogeneous, class A], _rational, _dAlembert]

Solve \begin {gather*} \boxed {\left (x^{3}+a y^{3}\right ) y^{\prime }-x^{2} y=0} \end {gather*}

Solution by Maple

Time used: 0.016 (sec). Leaf size: 23

dsolve((x^3+a*y(x)^3)*diff(y(x),x) = x^2*y(x),y(x), singsol=all)
 

\[ y \relax (x ) = \left (\frac {1}{a \LambertW \left (\frac {c_{1} x^{3}}{a}\right )}\right )^{\frac {1}{3}} x \]

Solution by Mathematica

Time used: 13.484 (sec). Leaf size: 113

DSolve[(x^3+a y[x]^3)y'[x]==x^2 y[x],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {x}{\sqrt [3]{a} \sqrt [3]{\text {ProductLog}\left (\frac {x^3 e^{-\frac {3 c_1}{a}}}{a}\right )}} \\ y(x)\to -\frac {\sqrt [3]{-1} x}{\sqrt [3]{a} \sqrt [3]{\text {ProductLog}\left (\frac {x^3 e^{-\frac {3 c_1}{a}}}{a}\right )}} \\ y(x)\to \frac {(-1)^{2/3} x}{\sqrt [3]{a} \sqrt [3]{\text {ProductLog}\left (\frac {x^3 e^{-\frac {3 c_1}{a}}}{a}\right )}} \\ y(x)\to 0 \\ \end{align*}