4.8.46 \(\sqrt {b^2+x^2} y'(x)=\sqrt {a^2+y(x)^2}\)

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

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica
cpu = 0.0556873 (sec), leaf count = 61

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

Maple
cpu = 0.014 (sec), leaf count = 33

\[ \left \{ \ln \left (x+\sqrt {{b}^{2}+{x}^{2}} \right ) -\ln \left (y \relax (x ) +\sqrt { \left (y \relax (x ) \right ) ^{2}+{a}^{2}} \right ) +{\it \_C1}=0 \right \} \] Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

ln(x+(b^2+x^2)^(1/2))-ln(y(x)+(y(x)^2+a^2)^(1/2))+_C1 = 0