4.19.40 \(x^2 \left (a^2-x^2\right ) y'(x)^2+1=0\)

ODE
\[ x^2 \left (a^2-x^2\right ) y'(x)^2+1=0 \] ODE Classification

[_quadrature]

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

Mathematica
cpu = 0.0533868 (sec), leaf count = 139

\[\left \{\left \{y(x)\to c_1-\frac {i x \sqrt {x^2-a^2} \log \left (\frac {2 \left (\sqrt {x^2-a^2}-i a\right )}{x}\right )}{a \sqrt {x^4-a^2 x^2}}\right \},\left \{y(x)\to c_1+\frac {i x \sqrt {x^2-a^2} \log \left (\frac {2 \left (\sqrt {x^2-a^2}-i a\right )}{x}\right )}{a \sqrt {x^4-a^2 x^2}}\right \}\right \}\]

Maple
cpu = 0.038 (sec), leaf count = 90

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

DSolve[1 + x^2*(a^2 - x^2)*y'[x]^2 == 0,y[x],x]

Mathematica raw output

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

Maple raw input

dsolve(x^2*(a^2-x^2)*diff(y(x),x)^2+1 = 0, y(x),'implicit')

Maple raw output

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