4.19.24 $(a^2-x^2)y^{\prime }(x{)}^2+x^2+2xy(x)y^{\prime }(x)=0$

ODE
$$(a^2-x^2)y^{\prime }(x{)}^2+x^2+2xy(x)y^{\prime }(x)=0$$ 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

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

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

$$\{{\left(y\left(x\right)\right)}^2-a^2+x^2=0,y\left(x\right)=\mathit{\_}\mathit{C}\mathit{1}{x}^2-\mathit{\_}\mathit{C}\mathit{1}{a}^2-\frac{1}{4\mathit{\_}\mathit{C}\mathit{1}}\}$$ 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