4.6.1 \(3 x y'(x)=y(x) \left (3 x y(x)^3 \log (x)+1\right )\)

ODE
\[ 3 x y'(x)=y(x) \left (3 x y(x)^3 \log (x)+1\right ) \] ODE Classification

[_Bernoulli]

Book solution method
The Bernoulli ODE

Mathematica
cpu = 0.0102411 (sec), leaf count = 115

\[\left \{\left \{y(x)\to \frac {(-2)^{2/3} \sqrt [3]{x}}{\sqrt [3]{4 c_1+3 x^2-6 x^2 \log (x)}}\right \},\left \{y(x)\to \frac {2^{2/3} \sqrt [3]{x}}{\sqrt [3]{4 c_1+3 x^2-6 x^2 \log (x)}}\right \},\left \{y(x)\to -\frac {\sqrt [3]{-1} 2^{2/3} \sqrt [3]{x}}{\sqrt [3]{4 c_1+3 x^2-6 x^2 \log (x)}}\right \}\right \}\]

Maple
cpu = 0.007 (sec), leaf count = 22

\[ \left \{ \left (y \relax (x ) \right ) ^{-3}+{\frac {3\,x\ln \relax (x ) }{2}}-{\frac {3\,x}{4}}-{\frac {{\it \_C1}}{x}}=0 \right \} \] Mathematica raw input

DSolve[3*x*y'[x] == y[x]*(1 + 3*x*Log[x]*y[x]^3),y[x],x]

Mathematica raw output

{{y[x] -> ((-2)^(2/3)*x^(1/3))/(3*x^2 + 4*C[1] - 6*x^2*Log[x])^(1/3)}, {y[x] -> 
(2^(2/3)*x^(1/3))/(3*x^2 + 4*C[1] - 6*x^2*Log[x])^(1/3)}, {y[x] -> -(((-1)^(1/3)
*2^(2/3)*x^(1/3))/(3*x^2 + 4*C[1] - 6*x^2*Log[x])^(1/3))}}

Maple raw input

dsolve(3*x*diff(y(x),x) = (1+3*x*y(x)^3*ln(x))*y(x), y(x),'implicit')

Maple raw output

1/y(x)^3+3/2*x*ln(x)-3/4*x-1/x*_C1 = 0