4.3.45 \(x y'(x)=\sqrt {a^2-x^2}\)

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

[_quadrature]

Book solution method
Separable ODE, Dependent variable missing

Mathematica
cpu = 0.0293514 (sec), leaf count = 46

\[\left \{\left \{y(x)\to \sqrt {a^2-x^2}-a \log \left (a \left (\sqrt {a^2-x^2}+a\right )\right )+a \log (x)+c_1\right \}\right \}\]

Maple
cpu = 0.017 (sec), leaf count = 56

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

DSolve[x*y'[x] == Sqrt[a^2 - x^2],y[x],x]

Mathematica raw output

{{y[x] -> Sqrt[a^2 - x^2] + C[1] + a*Log[x] - a*Log[a*(a + Sqrt[a^2 - x^2])]}}

Maple raw input

dsolve(x*diff(y(x),x) = (a^2-x^2)^(1/2), y(x),'implicit')

Maple raw output

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