##### 4.8.23 $$a^2+x^4 y'(x)+x^4 y(x)^2=0$$

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 = 0.246178 (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 = 0.138 (sec), leaf count = 34

$\left [y \left (x \right ) = -\frac {\sqrt {a^{2}}\, \tan \left (\frac {\sqrt {a^{2}}\, \left (x \textit {\_C1} -1\right )}{x}\right )-x}{x^{2}}\right ]$ 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))

Maple raw output

[y(x) = -((a^2)^(1/2)*tan((a^2)^(1/2)*(_C1*x-1)/x)-x)/x^2]