4.19.22 \(\left (a^2-x^2\right ) y'(x)^2=b^2\)

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

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