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
\[ \left \{ -{1\ln \left ( {\frac {1}{x} \left ( -2\,{a}^{2}+2\,\sqrt {-{a}^{2}}\sqrt {-{a}^{2}+{x}^{2}} \right ) } \right ) {\frac {1}{\sqrt {-{a}^{2}}}}}+{1\ln \left ( {\frac {1}{y \left ( x \right ) } \left ( -2\,{b}^{2}+2\,\sqrt {-{b}^{2}}\sqrt { \left ( y \left ( x \right ) \right ) ^{2}-{b}^{2}} \right ) } \right ) {\frac {1}{\sqrt {-{b}^{2}}}}}+{\it \_C1}=0 \right \} \] 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