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[Sec[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]