4.8.49 \(x \sqrt {x^2-a^2} y'(x)=y(x) \sqrt {y(x)^2-b^2}\)

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

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica
cpu = 0.0734998 (sec), leaf count = 100

\[\left \{\left \{y(x)\to -\frac {4 i b e^{i b c_1} \left (\frac {2 \sqrt {x^2-a^2}}{x}-\frac {2 i a}{x}\right )^{\frac {b}{a}}}{-4+e^{2 i b c_1} \left (\frac {2 \sqrt {x^2-a^2}}{x}-\frac {2 i a}{x}\right )^{\frac {2 b}{a}}}\right \}\right \}\]

Maple
cpu = 0.039 (sec), leaf count = 86

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

Maple raw input

dsolve(x*diff(y(x),x)*(-a^2+x^2)^(1/2) = y(x)*(y(x)^2-b^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)*(y(x)^2-b^2)^(1/2))/y(x))+_C1 = 0