4.15.6 \(x y'(x) \left (a+b x y(x)^3\right )+y(x) \left (a+c x^3 y(x)\right )=0\)

ODE
\[ x y'(x) \left (a+b x y(x)^3\right )+y(x) \left (a+c x^3 y(x)\right )=0 \] ODE Classification

[_rational]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.0417136 (sec), leaf count = 491

\[\left \{\left \{y(x)\to \frac {\sqrt [3]{3} \left (\sqrt {3} \sqrt {b^3 x^4 \left (27 a^2 b+x^2 \left (c x^2-2 c_1\right ){}^3\right )}+9 a b^2 x^2\right ){}^{2/3}+3^{2/3} b x^2 \left (2 c_1-c x^2\right )}{3 b x \sqrt [3]{\sqrt {3} \sqrt {b^3 x^4 \left (27 a^2 b+x^2 \left (c x^2-2 c_1\right ){}^3\right )}+9 a b^2 x^2}}\right \},\left \{y(x)\to \frac {i \sqrt [6]{3} \left (\sqrt {3}+i\right ) \left (\sqrt {3} \sqrt {b^3 x^4 \left (27 a^2 b+x^2 \left (c x^2-2 c_1\right ){}^3\right )}+9 a b^2 x^2\right ){}^{2/3}+\left (\sqrt {3}+3 i\right ) b x^2 \left (c x^2-2 c_1\right )}{2\ 3^{5/6} b x \sqrt [3]{\sqrt {3} \sqrt {b^3 x^4 \left (27 a^2 b+x^2 \left (c x^2-2 c_1\right ){}^3\right )}+9 a b^2 x^2}}\right \},\left \{y(x)\to \frac {\left (\sqrt {3}-3 i\right ) b x^2 \left (c x^2-2 c_1\right )-i \sqrt [6]{3} \left (\sqrt {3}-i\right ) \left (\sqrt {3} \sqrt {b^3 x^4 \left (27 a^2 b+x^2 \left (c x^2-2 c_1\right ){}^3\right )}+9 a b^2 x^2\right ){}^{2/3}}{2\ 3^{5/6} b x \sqrt [3]{\sqrt {3} \sqrt {b^3 x^4 \left (27 a^2 b+x^2 \left (c x^2-2 c_1\right ){}^3\right )}+9 a b^2 x^2}}\right \}\right \}\]

Maple
cpu = 0.114 (sec), leaf count = 27

\[ \left \{ -{\frac {c{x}^{2}}{2}}+{\frac {a}{xy \relax (x ) }}-{\frac {b \left (y \relax (x ) \right ) ^{2}}{2}}+{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[y[x]*(a + c*x^3*y[x]) + x*(a + b*x*y[x]^3)*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> (3^(1/3)*(9*a*b^2*x^2 + Sqrt[3]*Sqrt[b^3*x^4*(27*a^2*b + x^2*(c*x^2 - 
2*C[1])^3)])^(2/3) + 3^(2/3)*b*x^2*(-(c*x^2) + 2*C[1]))/(3*b*x*(9*a*b^2*x^2 + Sq
rt[3]*Sqrt[b^3*x^4*(27*a^2*b + x^2*(c*x^2 - 2*C[1])^3)])^(1/3))}, {y[x] -> (I*3^
(1/6)*(I + Sqrt[3])*(9*a*b^2*x^2 + Sqrt[3]*Sqrt[b^3*x^4*(27*a^2*b + x^2*(c*x^2 -
 2*C[1])^3)])^(2/3) + (3*I + Sqrt[3])*b*x^2*(c*x^2 - 2*C[1]))/(2*3^(5/6)*b*x*(9*
a*b^2*x^2 + Sqrt[3]*Sqrt[b^3*x^4*(27*a^2*b + x^2*(c*x^2 - 2*C[1])^3)])^(1/3))}, 
{y[x] -> ((-I)*3^(1/6)*(-I + Sqrt[3])*(9*a*b^2*x^2 + Sqrt[3]*Sqrt[b^3*x^4*(27*a^
2*b + x^2*(c*x^2 - 2*C[1])^3)])^(2/3) + (-3*I + Sqrt[3])*b*x^2*(c*x^2 - 2*C[1]))
/(2*3^(5/6)*b*x*(9*a*b^2*x^2 + Sqrt[3]*Sqrt[b^3*x^4*(27*a^2*b + x^2*(c*x^2 - 2*C
[1])^3)])^(1/3))}}

Maple raw input

dsolve(x*(a+b*x*y(x)^3)*diff(y(x),x)+(a+c*x^3*y(x))*y(x) = 0, y(x),'implicit')

Maple raw output

-1/2*c*x^2+1/y(x)/x*a-1/2*b*y(x)^2+_C1 = 0