##### 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 = 0.382972 (sec), leaf count = 84

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

Maple
cpu = 0.031 (sec), leaf count = 86

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

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))

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)*l
n((-2*b^2+2*(-b^2)^(1/2)*(y(x)^2-b^2)^(1/2))/y(x))+_C1 = 0]