ODE
\[ y(x) \left (a+x^2+y(x)^2\right ) y'(x)=x \left (a-x^2-y(x)^2\right ) \] ODE Classification
[_exact, _rational]
Book solution method
Exact equation
Mathematica ✓
cpu = 0.0244473 (sec), leaf count = 149
\[\left \{\left \{y(x)\to -\sqrt {-\sqrt {a^2+4 a x^2+4 c_1}-a-x^2}\right \},\left \{y(x)\to \sqrt {-\sqrt {a^2+4 a x^2+4 c_1}-a-x^2}\right \},\left \{y(x)\to -\sqrt {\sqrt {a^2+4 a x^2+4 c_1}-a-x^2}\right \},\left \{y(x)\to \sqrt {\sqrt {a^2+4 a x^2+4 c_1}-a-x^2}\right \}\right \}\]
Maple ✓
cpu = 0.016 (sec), leaf count = 37
\[ \left \{ {\frac { \left ( y \left ( x \right ) \right ) ^{4}}{4}}+{\frac { \left ( 2\,{x}^{2}+2\,a \right ) \left ( y \left ( x \right ) \right ) ^{2}}{4}}+{\frac {{x}^{4}}{4}}-{\frac {a{x}^{2}}{2}}+{\it \_C1}=0 \right \} \] Mathematica raw input
DSolve[y[x]*(a + x^2 + y[x]^2)*y'[x] == x*(a - x^2 - y[x]^2),y[x],x]
Mathematica raw output
{{y[x] -> -Sqrt[-a - x^2 - Sqrt[a^2 + 4*a*x^2 + 4*C[1]]]}, {y[x] -> Sqrt[-a - x^2 - Sqrt[a^2 + 4*a*x^2 + 4*C[1]]]}, {y[x] -> -Sqrt[-a - x^2 + Sqrt[a^2 + 4*a*x^2 + 4*C[1]]]}, {y[x] -> Sqrt[-a - x^2 + Sqrt[a^2 + 4*a*x^2 + 4*C[1]]]}}
Maple raw input
dsolve((a+x^2+y(x)^2)*y(x)*diff(y(x),x) = x*(a-x^2-y(x)^2), y(x),'implicit')
Maple raw output
1/4*y(x)^4+1/4*(2*x^2+2*a)*y(x)^2+1/4*x^4-1/2*a*x^2+_C1 = 0