4.19.24 \(\left (a^2-x^2\right ) y'(x)^2+x^2+2 x y(x) y'(x)=0\)

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

[[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

Book solution method
No Missing Variables ODE, Solve for \(y\)

Mathematica
cpu = 0.416386 (sec), leaf count = 26

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

Maple
cpu = 0.449 (sec), leaf count = 36

\[ \left \{ \left (y \relax (x ) \right ) ^{2}-{a}^{2}+{x}^{2}=0,y \relax (x ) ={\it \_C1}\,{x}^{2}-{\it \_C1}\,{a}^{2}-{\frac {1}{4\,{\it \_C1}}} \right \} \] Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

y(x)^2-a^2+x^2 = 0, y(x) = _C1*x^2-_C1*a^2-1/4/_C1