##### 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 = 0.178661 (sec), leaf count = 113

$\left \{\left \{y(x)\to -\frac {2 \sqrt {x} \sqrt {a-x^2} \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {5}{4};\frac {x^2}{a}\right )}{\sqrt {1-\frac {x^2}{a}}}+c_1\right \},\left \{y(x)\to \frac {2 \sqrt {x} \sqrt {a-x^2} \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {5}{4};\frac {x^2}{a}\right )}{\sqrt {1-\frac {x^2}{a}}}+c_1\right \}\right \}$

Maple
cpu = 0.059 (sec), leaf count = 46

$\left [y \left (x \right ) = \int \frac {\sqrt {x \left (-x^{2}+a \right )}}{x}d x +\textit {\_C1}, y \left (x \right ) = \int -\frac {\sqrt {x \left (-x^{2}+a \right )}}{x}d x +\textit {\_C1}\right ]$ Mathematica raw input

DSolve[x*y'[x]^2 == a - x^2,y[x],x]

Mathematica raw output

{{y[x] -> C[1] - (2*Sqrt[x]*Sqrt[a - x^2]*Hypergeometric2F1[-1/2, 1/4, 5/4, x^2/
a])/Sqrt[1 - x^2/a]}, {y[x] -> C[1] + (2*Sqrt[x]*Sqrt[a - x^2]*Hypergeometric2F1
[-1/2, 1/4, 5/4, x^2/a])/Sqrt[1 - x^2/a]}}

Maple raw input

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

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]