2.432   ODE No. 432

\[ \left (a+x y'(x)\right )^2-2 a y(x)+x^2=0 \] Mathematica : cpu = 1.13082 (sec), leaf count = 70


\[\text {Solve}\left [\left \{y(x)=\frac {2 a x K[1]+x^2 K[1]^2+a^2+x^2}{2 a},x=-\frac {a \sinh ^{-1}(K[1])}{\sqrt {K[1]^2+1}}+\frac {c_1}{\sqrt {K[1]^2+1}}\right \},\{y(x),K[1]\}\right ]\] Maple : cpu = 11.198 (sec), leaf count = 242


\[y \relax (x ) = \frac {-2 a \RootOf \left (\arcsinh \left (\textit {\_Z} \right )^{2} a^{2}-\textit {\_Z}^{2} x^{2}-2 \arcsinh \left (\textit {\_Z} \right ) c_{1} a +c_{1}^{2}-x^{2}\right ) \left (a \arcsinh \left (\RootOf \left (\arcsinh \left (\textit {\_Z} \right )^{2} a^{2}-\textit {\_Z}^{2} x^{2}-2 \arcsinh \left (\textit {\_Z} \right ) c_{1} a +c_{1}^{2}-x^{2}\right )\right )-c_{1}\right ) \sqrt {\RootOf \left (\arcsinh \left (\textit {\_Z} \right )^{2} a^{2}-\textit {\_Z}^{2} x^{2}-2 \arcsinh \left (\textit {\_Z} \right ) c_{1} a +c_{1}^{2}-x^{2}\right )^{2}+1}+\left (\RootOf \left (\arcsinh \left (\textit {\_Z} \right )^{2} a^{2}-\textit {\_Z}^{2} x^{2}-2 \arcsinh \left (\textit {\_Z} \right ) c_{1} a +c_{1}^{2}-x^{2}\right )^{2}+1\right ) \left (\RootOf \left (\arcsinh \left (\textit {\_Z} \right )^{2} a^{2}-\textit {\_Z}^{2} x^{2}-2 \arcsinh \left (\textit {\_Z} \right ) c_{1} a +c_{1}^{2}-x^{2}\right )^{2} x^{2}+a^{2}+x^{2}\right )}{2 a \left (\RootOf \left (\arcsinh \left (\textit {\_Z} \right )^{2} a^{2}-\textit {\_Z}^{2} x^{2}-2 \arcsinh \left (\textit {\_Z} \right ) c_{1} a +c_{1}^{2}-x^{2}\right )^{2}+1\right )}\]