ODE
\[ \left (a^2-x^2\right ) y'(x)^2=b^2 \] ODE Classification
[_quadrature]
Book solution method
Missing Variables ODE, Dependent variable missing, Solve for \(y'\)
Mathematica ✓
cpu = 0.0225286 (sec), leaf count = 52
\[\left \{\left \{y(x)\to b \tan ^{-1}\left (\frac {x}{\sqrt {a^2-x^2}}\right )+c_1\right \},\left \{y(x)\to c_1-b \tan ^{-1}\left (\frac {x}{\sqrt {a^2-x^2}}\right )\right \}\right \}\]
Maple ✓
cpu = 0.035 (sec), leaf count = 44
\[ \left \{ y \left ( x \right ) =-b\arctan \left ( {x{\frac {1}{\sqrt {{a}^{2}-{x}^{2}}}}} \right ) +{\it \_C1},y \left ( x \right ) =b\arctan \left ( {x{\frac {1}{\sqrt {{a}^{2}-{x}^{2}}}}} \right ) +{\it \_C1} \right \} \] Mathematica raw input
DSolve[(a^2 - x^2)*y'[x]^2 == b^2,y[x],x]
Mathematica raw output
{{y[x] -> b*ArcTan[x/Sqrt[a^2 - x^2]] + C[1]}, {y[x] -> -(b*ArcTan[x/Sqrt[a^2 - x^2]]) + C[1]}}
Maple raw input
dsolve((a^2-x^2)*diff(y(x),x)^2 = b^2, y(x),'implicit')
Maple raw output
y(x) = b*arctan(x/(a^2-x^2)^(1/2))+_C1, y(x) = -b*arctan(x/(a^2-x^2)^(1/2))+_C1