##### 4.8.47 $$\sqrt {b^2-x^2} y'(x)=\sqrt {a^2-y(x)^2}$$

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

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica
cpu = 0.417756 (sec), leaf count = 106

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

Maple
cpu = 0.02 (sec), leaf count = 37

$\left [\arctan \left (\frac {x}{\sqrt {b^{2}-x^{2}}}\right )-\arctan \left (\frac {y \left (x \right )}{\sqrt {a^{2}-y \left (x \right )^{2}}}\right )+\textit {\_C1} = 0\right ]$ Mathematica raw input

DSolve[Sqrt[b^2 - x^2]*y'[x] == Sqrt[a^2 - y[x]^2],y[x],x]

Mathematica raw output

{{y[x] -> -((a*Tan[ArcTan[x/Sqrt[b^2 - x^2]] + C[1]])/Sqrt[Sec[ArcTan[x/Sqrt[b^2
 - x^2]] + C[1]]^2])}, {y[x] -> (a*Tan[ArcTan[x/Sqrt[b^2 - x^2]] + C[1]])/Sqrt[S
ec[ArcTan[x/Sqrt[b^2 - x^2]] + C[1]]^2]}}

Maple raw input

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

Maple raw output

[arctan(x/(b^2-x^2)^(1/2))-arctan(y(x)/(a^2-y(x)^2)^(1/2))+_C1 = 0]