##### 4.6.27 $$x^2 y'(x)=y(x) \left (a x+b y(x)^3\right )$$

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

[[_homogeneous, class G], _rational, _Bernoulli]

Book solution method
The Bernoulli ODE

Mathematica
cpu = 0.254109 (sec), leaf count = 144

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

Maple
cpu = 0.054 (sec), leaf count = 344

$\left [y \left (x \right ) = \frac {\left (x \left (3 a -1\right ) \left (3 \textit {\_C1} a \,x^{-3 a +1}-\textit {\_C1} \,x^{-3 a +1}-3 b \right )^{2}\right )^{\frac {1}{3}}}{3 \textit {\_C1} a \,x^{-3 a +1}-\textit {\_C1} \,x^{-3 a +1}-3 b}, y \left (x \right ) = -\frac {\left (x \left (3 a -1\right ) \left (3 \textit {\_C1} a \,x^{-3 a +1}-\textit {\_C1} \,x^{-3 a +1}-3 b \right )^{2}\right )^{\frac {1}{3}}}{2 \left (3 \textit {\_C1} a \,x^{-3 a +1}-\textit {\_C1} \,x^{-3 a +1}-3 b \right )}-\frac {i \sqrt {3}\, \left (x \left (3 a -1\right ) \left (3 \textit {\_C1} a \,x^{-3 a +1}-\textit {\_C1} \,x^{-3 a +1}-3 b \right )^{2}\right )^{\frac {1}{3}}}{2 \left (3 \textit {\_C1} a \,x^{-3 a +1}-\textit {\_C1} \,x^{-3 a +1}-3 b \right )}, y \left (x \right ) = -\frac {\left (x \left (3 a -1\right ) \left (3 \textit {\_C1} a \,x^{-3 a +1}-\textit {\_C1} \,x^{-3 a +1}-3 b \right )^{2}\right )^{\frac {1}{3}}}{2 \left (3 \textit {\_C1} a \,x^{-3 a +1}-\textit {\_C1} \,x^{-3 a +1}-3 b \right )}+\frac {i \sqrt {3}\, \left (x \left (3 a -1\right ) \left (3 \textit {\_C1} a \,x^{-3 a +1}-\textit {\_C1} \,x^{-3 a +1}-3 b \right )^{2}\right )^{\frac {1}{3}}}{6 \textit {\_C1} a \,x^{-3 a +1}-2 \textit {\_C1} \,x^{-3 a +1}-6 b}\right ]$ Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

[y(x) = 1/(3*_C1*a*x^(-3*a+1)-_C1*x^(-3*a+1)-3*b)*(x*(3*a-1)*(3*_C1*a*x^(-3*a+1)
-_C1*x^(-3*a+1)-3*b)^2)^(1/3), y(x) = -1/2/(3*_C1*a*x^(-3*a+1)-_C1*x^(-3*a+1)-3*
b)*(x*(3*a-1)*(3*_C1*a*x^(-3*a+1)-_C1*x^(-3*a+1)-3*b)^2)^(1/3)-1/2*I*3^(1/2)/(3*
_C1*a*x^(-3*a+1)-_C1*x^(-3*a+1)-3*b)*(x*(3*a-1)*(3*_C1*a*x^(-3*a+1)-_C1*x^(-3*a+
1)-3*b)^2)^(1/3), y(x) = -1/2/(3*_C1*a*x^(-3*a+1)-_C1*x^(-3*a+1)-3*b)*(x*(3*a-1)
*(3*_C1*a*x^(-3*a+1)-_C1*x^(-3*a+1)-3*b)^2)^(1/3)+1/2*I*3^(1/2)/(3*_C1*a*x^(-3*a
+1)-_C1*x^(-3*a+1)-3*b)*(x*(3*a-1)*(3*_C1*a*x^(-3*a+1)-_C1*x^(-3*a+1)-3*b)^2)^(1
/3)]