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