4.13.31 \(3 \left (x^2-y(x)^2\right ) y'(x)-2 y(x)^3+6 x (x+1) y(x)+3 e^x=0\)

ODE
\[ 3 \left (x^2-y(x)^2\right ) y'(x)-2 y(x)^3+6 x (x+1) y(x)+3 e^x=0 \] ODE Classification

[`y=_G(x,y')`]

Book solution method
Change of Variable, new dependent variable

Mathematica
cpu = 0.0490614 (sec), leaf count = 501

\[\left \{\left \{y(x)\to -\frac {e^{-2 x} \sqrt [3]{\sqrt {e^{8 x} \left (-2 c_1 e^{3 x}+c_1^2-4 e^{4 x} x^6+e^{6 x}\right )}+c_1 e^{4 x}-e^{7 x}}}{\sqrt [3]{2}}-\frac {\sqrt [3]{2} e^{2 x} x^2}{\sqrt [3]{\sqrt {e^{8 x} \left (-2 c_1 e^{3 x}+c_1^2-4 e^{4 x} x^6+e^{6 x}\right )}+c_1 e^{4 x}-e^{7 x}}}\right \},\left \{y(x)\to \frac {e^{-2 x} \left (\sqrt [3]{2} \left (1-i \sqrt {3}\right ) \left (\sqrt {e^{8 x} \left (-2 c_1 e^{3 x}+c_1^2-4 e^{4 x} x^6+e^{6 x}\right )}+c_1 e^{4 x}-e^{7 x}\right ){}^{2/3}+2 \left (1+i \sqrt {3}\right ) e^{4 x} x^2\right )}{2\ 2^{2/3} \sqrt [3]{\sqrt {e^{8 x} \left (-2 c_1 e^{3 x}+c_1^2-4 e^{4 x} x^6+e^{6 x}\right )}+c_1 e^{4 x}-e^{7 x}}}\right \},\left \{y(x)\to \frac {e^{-2 x} \left (\sqrt [3]{2} \left (1+i \sqrt {3}\right ) \left (\sqrt {e^{8 x} \left (-2 c_1 e^{3 x}+c_1^2-4 e^{4 x} x^6+e^{6 x}\right )}+c_1 e^{4 x}-e^{7 x}\right ){}^{2/3}+2 \left (1-i \sqrt {3}\right ) e^{4 x} x^2\right )}{2\ 2^{2/3} \sqrt [3]{\sqrt {e^{8 x} \left (-2 c_1 e^{3 x}+c_1^2-4 e^{4 x} x^6+e^{6 x}\right )}+c_1 e^{4 x}-e^{7 x}}}\right \}\right \}\]

Maple
cpu = 0.046 (sec), leaf count = 30

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

DSolve[3*E^x + 6*x*(1 + x)*y[x] - 2*y[x]^3 + 3*(x^2 - y[x]^2)*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> -((2^(1/3)*E^(2*x)*x^2)/(-E^(7*x) + E^(4*x)*C[1] + Sqrt[E^(8*x)*(E^(6*
x) - 4*E^(4*x)*x^6 - 2*E^(3*x)*C[1] + C[1]^2)])^(1/3)) - (-E^(7*x) + E^(4*x)*C[1
] + Sqrt[E^(8*x)*(E^(6*x) - 4*E^(4*x)*x^6 - 2*E^(3*x)*C[1] + C[1]^2)])^(1/3)/(2^
(1/3)*E^(2*x))}, {y[x] -> (2*(1 + I*Sqrt[3])*E^(4*x)*x^2 + 2^(1/3)*(1 - I*Sqrt[3
])*(-E^(7*x) + E^(4*x)*C[1] + Sqrt[E^(8*x)*(E^(6*x) - 4*E^(4*x)*x^6 - 2*E^(3*x)*
C[1] + C[1]^2)])^(2/3))/(2*2^(2/3)*E^(2*x)*(-E^(7*x) + E^(4*x)*C[1] + Sqrt[E^(8*
x)*(E^(6*x) - 4*E^(4*x)*x^6 - 2*E^(3*x)*C[1] + C[1]^2)])^(1/3))}, {y[x] -> (2*(1
 - I*Sqrt[3])*E^(4*x)*x^2 + 2^(1/3)*(1 + I*Sqrt[3])*(-E^(7*x) + E^(4*x)*C[1] + S
qrt[E^(8*x)*(E^(6*x) - 4*E^(4*x)*x^6 - 2*E^(3*x)*C[1] + C[1]^2)])^(2/3))/(2*2^(2
/3)*E^(2*x)*(-E^(7*x) + E^(4*x)*C[1] + Sqrt[E^(8*x)*(E^(6*x) - 4*E^(4*x)*x^6 - 2
*E^(3*x)*C[1] + C[1]^2)])^(1/3))}}

Maple raw input

dsolve(3*(x^2-y(x)^2)*diff(y(x),x)+3*exp(x)+6*x*y(x)*(1+x)-2*y(x)^3 = 0, y(x),'implicit')

Maple raw output

3*exp(2*x)*x^2*y(x)-exp(2*x)*y(x)^3+exp(3*x)+_C1 = 0