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]