4.15.11 \(\left (a^2 x^2+\left (x^2+y(x)^2\right )^2\right ) y'(x)=a^2 x y(x)\)

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

[_rational]

Book solution method
Change of Variable, Two new variables

Mathematica
cpu = 0.153113 (sec), leaf count = 239

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

Maple
cpu = 0.422 (sec), leaf count = 37

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

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

(y(x)^4+(a^2+x^2+_C1)*y(x)^2+x^2*_C1)/(x^2+y(x)^2) = 0