4.9.33 $y(x)y^{\prime }(x)=\sqrt{y(x{)}^2-a^2}$

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

[_quadrature]

Book solution method
Separable ODE, Independent variable missing

Mathematica ✓
cpu = 0.0161185 (sec), leaf count = 39

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

Maple ✓
cpu = 0.012 (sec), leaf count = 29

$$\{x+(a-y\left(x\right))(a+y\left(x\right))\frac{1}{\sqrt{{\left(y\left(x\right)\right)}^2-a^2}}+\mathit{\_}\mathit{C}\mathit{1}=0\}$$ 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+(a-y(x))*(a+y(x))/(y(x)^2-a^2)^(1/2)+_C1 = 0