4.17.48 \(x y'(x)^2=a-x^2\)

ODE
\[ x y'(x)^2=a-x^2 \] ODE Classification

[_quadrature]

Book solution method
Missing Variables ODE, Dependent variable missing, Solve for \(y'\)

Mathematica
cpu = 0.236688 (sec), leaf count = 167

\[\left \{\left \{y(x)\to c_1-\frac {2 \sqrt {a-x^2} \left (x^{3/2}-\frac {2 i a F\left (\left .i \sinh ^{-1}\left (\frac {\sqrt {-\sqrt {a}}}{\sqrt {x}}\right )\right |-1\right )}{\sqrt {-\sqrt {a}} \sqrt {1-\frac {a}{x^2}}}\right )}{3 x}\right \},\left \{y(x)\to c_1+\frac {2 \sqrt {a-x^2} \left (x^{3/2}-\frac {2 i a F\left (\left .i \sinh ^{-1}\left (\frac {\sqrt {-\sqrt {a}}}{\sqrt {x}}\right )\right |-1\right )}{\sqrt {-\sqrt {a}} \sqrt {1-\frac {a}{x^2}}}\right )}{3 x}\right \}\right \}\]

Maple
cpu = 0.025 (sec), leaf count = 46

\[ \left \{ y \relax (x ) =\int \!{\frac {1}{x}\sqrt {x \left (-{x}^{2}+a \right ) }}\,{\rm d}x+{\it \_C1},y \relax (x ) =\int \!-{\frac {1}{x}\sqrt {x \left (-{x}^{2}+a \right ) }}\,{\rm d}x+{\it \_C1} \right \} \] Mathematica raw input

DSolve[x*y'[x]^2 == a - x^2,y[x],x]

Mathematica raw output

{{y[x] -> C[1] - (2*Sqrt[a - x^2]*(x^(3/2) - ((2*I)*a*EllipticF[I*ArcSinh[Sqrt[-
Sqrt[a]]/Sqrt[x]], -1])/(Sqrt[-Sqrt[a]]*Sqrt[1 - a/x^2])))/(3*x)}, {y[x] -> C[1]
 + (2*Sqrt[a - x^2]*(x^(3/2) - ((2*I)*a*EllipticF[I*ArcSinh[Sqrt[-Sqrt[a]]/Sqrt[
x]], -1])/(Sqrt[-Sqrt[a]]*Sqrt[1 - a/x^2])))/(3*x)}}

Maple raw input

dsolve(x*diff(y(x),x)^2 = -x^2+a, y(x),'implicit')

Maple raw output

y(x) = Int(1/x*(x*(-x^2+a))^(1/2),x)+_C1, y(x) = Int(-1/x*(x*(-x^2+a))^(1/2),x)+
_C1