4.9.32 $y(x)y^{\prime }(x)=\sqrt{a^2+y(x{)}^2}$

ODE
$$y(x)y^{\prime }(x)=\sqrt{a^2+y(x{)}^2}$$ ODE Classification

[_quadrature]

Book solution method
Separable ODE, Independent variable missing

Mathematica ✓
cpu = 0.0143621 (sec), leaf count = 43

$$\{\{y(x)\to -\sqrt{(c_1+x){}^2-a^2}\},\{y(x)\to \sqrt{(c_1+x){}^2-a^2}\}\}$$

Maple ✓
cpu = 0.009 (sec), leaf count = 18

$$\{x-\sqrt{{\left(y\left(x\right)\right)}^2+a^2}+\mathit{\_}\mathit{C}\mathit{1}=0\}$$ Mathematica raw input

DSolve[y[x]*y'[x] == Sqrt[a^2 + y[x]^2],y[x],x]

Mathematica raw output

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

Maple raw input

dsolve(y(x)*diff(y(x),x) = (y(x)^2+a^2)^(1/2), y(x),'implicit')

Maple raw output

x-(y(x)^2+a^2)^(1/2)+_C1 = 0