4.14.42 \(\left (a y(x)^3+x^3\right ) y'(x)=x^2 y(x)\)

ODE
\[ \left (a y(x)^3+x^3\right ) y'(x)=x^2 y(x) \] ODE Classification

[[_homogeneous, `class A`], _rational, _dAlembert]

Book solution method
Homogeneous equation

Mathematica
cpu = 0.0378831 (sec), leaf count = 108

\[\left \{\left \{y(x)\to \frac {x}{\sqrt [3]{a} \sqrt [3]{W\left (\frac {x^3 e^{-\frac {3 c_1}{a}}}{a}\right )}}\right \},\left \{y(x)\to -\frac {\sqrt [3]{-1} x}{\sqrt [3]{a} \sqrt [3]{W\left (\frac {x^3 e^{-\frac {3 c_1}{a}}}{a}\right )}}\right \},\left \{y(x)\to \frac {(-1)^{2/3} x}{\sqrt [3]{a} \sqrt [3]{W\left (\frac {x^3 e^{-\frac {3 c_1}{a}}}{a}\right )}}\right \}\right \}\]

Maple
cpu = 0.015 (sec), leaf count = 32

\[ \left \{ -\ln \left ({\frac {y \relax (x ) }{x}} \right ) +{\frac {{x}^{3}}{3\,a \left (y \relax (x ) \right ) ^{3}}}-\ln \relax (x ) -{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[(x^3 + a*y[x]^3)*y'[x] == x^2*y[x],y[x],x]

Mathematica raw output

{{y[x] -> x/(a^(1/3)*ProductLog[x^3/(a*E^((3*C[1])/a))]^(1/3))}, {y[x] -> -(((-1
)^(1/3)*x)/(a^(1/3)*ProductLog[x^3/(a*E^((3*C[1])/a))]^(1/3)))}, {y[x] -> ((-1)^
(2/3)*x)/(a^(1/3)*ProductLog[x^3/(a*E^((3*C[1])/a))]^(1/3))}}

Maple raw input

dsolve((x^3+a*y(x)^3)*diff(y(x),x) = x^2*y(x), y(x),'implicit')

Maple raw output

-ln(y(x)/x)+1/3/a*x^3/y(x)^3-ln(x)-_C1 = 0