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 = 0.048327 (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 = 0.018 (sec), leaf count = 37

\[ \left \{ \arctan \left ({x{\frac {1}{\sqrt {{b}^{2}-{x}^{2}}}}} \right ) -\arctan \left ({y \relax (x ) {\frac {1}{\sqrt {{a}^{2}- \left (y \relax (x ) \right ) ^{2}}}}} \right ) +{\it \_C1}=0 \right \} \] 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[Se
c[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),'implicit')

Maple raw output

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