ODE
\[ a^2+x^4 y'(x)+x^4 y(x)^2=0 \] ODE Classification
[_rational, [_Riccati, _special]]
Book solution method
Riccati ODE, Main form
Mathematica ✓
cpu = 0.0116087 (sec), leaf count = 112
\[\left \{\left \{y(x)\to \frac {\left (x+i \sqrt {-a^2} c_1\right ) \cosh \left (\frac {\sqrt {-a^2}}{x}\right )-\left (\sqrt {-a^2}+i c_1 x\right ) \sinh \left (\frac {\sqrt {-a^2}}{x}\right )}{x^2 \left (\cosh \left (\frac {\sqrt {-a^2}}{x}\right )-i c_1 \sinh \left (\frac {\sqrt {-a^2}}{x}\right )\right )}\right \}\right \}\]
Maple ✓
cpu = 0.079 (sec), leaf count = 34
\[ \left \{ -{x}^{2}y \left ( x \right ) -\tan \left ( {\frac {{\it \_C1}\,x-1}{x}\sqrt {{a}^{2}}} \right ) \sqrt {{a}^{2}}+x=0 \right \} \] Mathematica raw input
DSolve[a^2 + x^4*y[x]^2 + x^4*y'[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> ((x + I*Sqrt[-a^2]*C[1])*Cosh[Sqrt[-a^2]/x] - (Sqrt[-a^2] + I*x*C[1])*Sinh[Sqrt[-a^2]/x])/(x^2*(Cosh[Sqrt[-a^2]/x] - I*C[1]*Sinh[Sqrt[-a^2]/x]))}}
Maple raw input
dsolve(x^4*diff(y(x),x)+a^2+x^4*y(x)^2 = 0, y(x),'implicit')
Maple raw output
-x^2*y(x)-tan((a^2)^(1/2)*(_C1*x-1)/x)*(a^2)^(1/2)+x = 0