##### 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 = 0.340584 (sec), leaf count = 149

$\left \{\left \{y(x)\to -\frac {e^{-c_1} \sqrt {a^2 \left (2 x \left (\left (-1+e^{4 c_1}\right ) \sqrt {b^2+x^2}+\left (1+e^{4 c_1}\right ) x\right )+b^2 \left (-1+e^{2 c_1}\right ){}^2\right )}}{2 b}\right \},\left \{y(x)\to \frac {e^{-c_1} \sqrt {a^2 \left (2 x \left (\left (-1+e^{4 c_1}\right ) \sqrt {b^2+x^2}+\left (1+e^{4 c_1}\right ) x\right )+b^2 \left (-1+e^{2 c_1}\right ){}^2\right )}}{2 b}\right \}\right \}$

Maple
cpu = 0.016 (sec), leaf count = 33

$\left [\ln \left (x +\sqrt {b^{2}+x^{2}}\right )-\ln \left (y \left (x \right )+\sqrt {y \left (x \right )^{2}+a^{2}}\right )+\textit {\_C1} = 0\right ]$ 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] -> -1/2*Sqrt[a^2*(b^2*(-1 + E^(2*C[1]))^2 + 2*x*((1 + E^(4*C[1]))*x + (-1
 + E^(4*C[1]))*Sqrt[b^2 + x^2]))]/(b*E^C[1])}, {y[x] -> Sqrt[a^2*(b^2*(-1 + E^(2
*C[1]))^2 + 2*x*((1 + E^(4*C[1]))*x + (-1 + E^(4*C[1]))*Sqrt[b^2 + x^2]))]/(2*b*
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))

Maple raw output

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