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 = 0.0949498 (sec), leaf count = 90

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

Maple
cpu = 0.036 (sec), leaf count = 74

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

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),'implicit')

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