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

ODE
\[ y(x) y'(x)=\sqrt {y(x)^2-a^2} \] ODE Classification

[_quadrature]

Book solution method
Separable ODE, Independent variable missing

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

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

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

\[ \left \{ x+{ \left (a-y \relax (x ) \right ) \left (a+y \relax (x ) \right ) {\frac {1}{\sqrt { \left (y \relax (x ) \right ) ^{2}-{a}^{2}}}}}+{\it \_C1}=0 \right \} \] 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