4.13.8 \(\left (a^2+x^2+y(x)^2\right ) y'(x)+b^2+x^2+2 x y(x)=0\)

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

[_exact, _rational]

Book solution method
Exact equation

Mathematica
cpu = 0.0293924 (sec), leaf count = 438

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

Maple
cpu = 0.021 (sec), leaf count = 33

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

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

Mathematica raw output

{{y[x] -> (-2*a^2 - 2*x^2 + 2^(1/3)*(-3*b^2*x - x^3 + Sqrt[4*(a^2 + x^2)^3 + (3*
b^2*x + x^3 - 3*C[1])^2] + 3*C[1])^(2/3))/(2^(2/3)*(-3*b^2*x - x^3 + Sqrt[4*(a^2
 + x^2)^3 + (3*b^2*x + x^3 - 3*C[1])^2] + 3*C[1])^(1/3))}, {y[x] -> ((1 + I*Sqrt
[3])*(a^2 + x^2))/(2^(2/3)*(-3*b^2*x - x^3 + Sqrt[4*(a^2 + x^2)^3 + (3*b^2*x + x
^3 - 3*C[1])^2] + 3*C[1])^(1/3)) + ((I/2)*(I + Sqrt[3])*(-3*b^2*x - x^3 + Sqrt[4
*(a^2 + x^2)^3 + (3*b^2*x + x^3 - 3*C[1])^2] + 3*C[1])^(1/3))/2^(1/3)}, {y[x] ->
 ((1 - I*Sqrt[3])*(a^2 + x^2))/(2^(2/3)*(-3*b^2*x - x^3 + Sqrt[4*(a^2 + x^2)^3 +
 (3*b^2*x + x^3 - 3*C[1])^2] + 3*C[1])^(1/3)) - ((I/2)*(-I + Sqrt[3])*(-3*b^2*x 
- x^3 + Sqrt[4*(a^2 + x^2)^3 + (3*b^2*x + x^3 - 3*C[1])^2] + 3*C[1])^(1/3))/2^(1
/3)}}

Maple raw input

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

Maple raw output

b^2*x+1/3*x^3+x^2*y(x)+a^2*y(x)+1/3*y(x)^3+_C1 = 0