4.13.5 \(\left (x^2+y(x)^2\right ) y'(x)+2 x (y(x)+2 x)=0\)

ODE
\[ \left (x^2+y(x)^2\right ) y'(x)+2 x (y(x)+2 x)=0 \] ODE Classification

[[_homogeneous, `class A`], _exact, _rational, _dAlembert]

Book solution method
Exact equation

Mathematica
cpu = 0.226608 (sec), leaf count = 372

\[\left \{\left \{y(x)\to \frac {\sqrt [3]{\sqrt {-8 e^{3 c_1} x^3+e^{6 c_1}+20 x^6}+e^{3 c_1}-4 x^3}}{\sqrt [3]{2}}-\frac {\sqrt [3]{2} x^2}{\sqrt [3]{\sqrt {-8 e^{3 c_1} x^3+e^{6 c_1}+20 x^6}+e^{3 c_1}-4 x^3}}\right \},\left \{y(x)\to \frac {i \sqrt [3]{2} \left (\sqrt {3}+i\right ) \left (\sqrt {-8 e^{3 c_1} x^3+e^{6 c_1}+20 x^6}+e^{3 c_1}-4 x^3\right ){}^{2/3}+\left (2+2 i \sqrt {3}\right ) x^2}{2\ 2^{2/3} \sqrt [3]{\sqrt {-8 e^{3 c_1} x^3+e^{6 c_1}+20 x^6}+e^{3 c_1}-4 x^3}}\right \},\left \{y(x)\to \frac {\left (1-i \sqrt {3}\right ) x^2}{2^{2/3} \sqrt [3]{\sqrt {-8 e^{3 c_1} x^3+e^{6 c_1}+20 x^6}+e^{3 c_1}-4 x^3}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {-8 e^{3 c_1} x^3+e^{6 c_1}+20 x^6}+e^{3 c_1}-4 x^3}}{2 \sqrt [3]{2}}\right \}\right \}\]

Maple
cpu = 0.013 (sec), leaf count = 35

\[ \left \{ -{\frac {1}{3}\ln \left ({\frac {4\,{x}^{3}+3\,{x}^{2}y \relax (x ) + \left (y \relax (x ) \right ) ^{3}}{{x}^{3}}} \right ) }-\ln \relax (x ) -{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[2*x*(2*x + y[x]) + (x^2 + y[x]^2)*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> -((2^(1/3)*x^2)/(E^(3*C[1]) - 4*x^3 + Sqrt[E^(6*C[1]) - 8*E^(3*C[1])*x
^3 + 20*x^6])^(1/3)) + (E^(3*C[1]) - 4*x^3 + Sqrt[E^(6*C[1]) - 8*E^(3*C[1])*x^3 
+ 20*x^6])^(1/3)/2^(1/3)}, {y[x] -> ((2 + (2*I)*Sqrt[3])*x^2 + I*2^(1/3)*(I + Sq
rt[3])*(E^(3*C[1]) - 4*x^3 + Sqrt[E^(6*C[1]) - 8*E^(3*C[1])*x^3 + 20*x^6])^(2/3)
)/(2*2^(2/3)*(E^(3*C[1]) - 4*x^3 + Sqrt[E^(6*C[1]) - 8*E^(3*C[1])*x^3 + 20*x^6])
^(1/3))}, {y[x] -> ((1 - I*Sqrt[3])*x^2)/(2^(2/3)*(E^(3*C[1]) - 4*x^3 + Sqrt[E^(
6*C[1]) - 8*E^(3*C[1])*x^3 + 20*x^6])^(1/3)) - ((1 + I*Sqrt[3])*(E^(3*C[1]) - 4*
x^3 + Sqrt[E^(6*C[1]) - 8*E^(3*C[1])*x^3 + 20*x^6])^(1/3))/(2*2^(1/3))}}

Maple raw input

dsolve((x^2+y(x)^2)*diff(y(x),x)+2*x*(2*x+y(x)) = 0, y(x),'implicit')

Maple raw output

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