##### 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 = 0.250612 (sec), leaf count = 115

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

Maple
cpu = 0.04 (sec), leaf count = 234

$\left [y \left (x \right ) = \frac {\left (-4 x \left (6 x^{2} \ln \left (x \right )-3 x^{2}-4 \textit {\_C1} \right )^{2}\right )^{\frac {1}{3}}}{6 x^{2} \ln \left (x \right )-3 x^{2}-4 \textit {\_C1}}, y \left (x \right ) = -\frac {\left (-4 x \left (6 x^{2} \ln \left (x \right )-3 x^{2}-4 \textit {\_C1} \right )^{2}\right )^{\frac {1}{3}}}{2 \left (6 x^{2} \ln \left (x \right )-3 x^{2}-4 \textit {\_C1} \right )}-\frac {i \sqrt {3}\, \left (-4 x \left (6 x^{2} \ln \left (x \right )-3 x^{2}-4 \textit {\_C1} \right )^{2}\right )^{\frac {1}{3}}}{2 \left (6 x^{2} \ln \left (x \right )-3 x^{2}-4 \textit {\_C1} \right )}, y \left (x \right ) = -\frac {\left (-4 x \left (6 x^{2} \ln \left (x \right )-3 x^{2}-4 \textit {\_C1} \right )^{2}\right )^{\frac {1}{3}}}{2 \left (6 x^{2} \ln \left (x \right )-3 x^{2}-4 \textit {\_C1} \right )}+\frac {i \sqrt {3}\, \left (-4 x \left (6 x^{2} \ln \left (x \right )-3 x^{2}-4 \textit {\_C1} \right )^{2}\right )^{\frac {1}{3}}}{12 x^{2} \ln \left (x \right )-6 x^{2}-8 \textit {\_C1}}\right ]$ 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))

Maple raw output

[y(x) = 1/(6*x^2*ln(x)-3*x^2-4*_C1)*(-4*x*(6*x^2*ln(x)-3*x^2-4*_C1)^2)^(1/3), y(
x) = -1/2/(6*x^2*ln(x)-3*x^2-4*_C1)*(-4*x*(6*x^2*ln(x)-3*x^2-4*_C1)^2)^(1/3)-1/2
*I*3^(1/2)/(6*x^2*ln(x)-3*x^2-4*_C1)*(-4*x*(6*x^2*ln(x)-3*x^2-4*_C1)^2)^(1/3), y
(x) = -1/2/(6*x^2*ln(x)-3*x^2-4*_C1)*(-4*x*(6*x^2*ln(x)-3*x^2-4*_C1)^2)^(1/3)+1/
2*I*3^(1/2)/(6*x^2*ln(x)-3*x^2-4*_C1)*(-4*x*(6*x^2*ln(x)-3*x^2-4*_C1)^2)^(1/3)]