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

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

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

Book solution method
Exact equation

Mathematica
cpu = 0.602625 (sec), leaf count = 359

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

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

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

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

Maple raw input

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

Maple raw output

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