4.8.48 $$x \sqrt {a^2+x^2} y'(x)=y(x) \sqrt {b^2+y(x)^2}$$

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

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica
cpu = 0.364407 (sec), leaf count = 84

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

Maple
cpu = 0.032 (sec), leaf count = 74

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

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

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