#### 2.432   ODE No. 432

$\left (a+x y'(x)\right )^2-2 a y(x)+x^2=0$ Mathematica : cpu = 1.15123 (sec), leaf count = 64

$\text {Solve}\left [\left \{y(x)=\frac {a^2+2 a \text {K\365773} x+\text {K\365773}^2 x^2+x^2}{2 a},x=\frac {c_1}{\sqrt {\text {K\365773}^2+1}}-\frac {a \sinh ^{-1}(\text {K\365773})}{\sqrt {\text {K\365773}^2+1}}\right \},\{y(x),\text {K\365773}\}\right ]$ Maple : cpu = 7.73 (sec), leaf count = 242

$\left \{ y \left ( x \right ) ={\frac {1}{2\,a \left ( \left ( {\it RootOf} \left ( \left ( {\it Arcsinh} \left ( {\it \_Z} \right ) \right ) ^{2}{a}^{2}-{{\it \_Z}}^{2}{x}^{2}-2\,{\it Arcsinh} \left ( {\it \_Z} \right ) {\it \_C1}\,a+{{\it \_C1}}^{2}-{x}^{2} \right ) \right ) ^{2}+1 \right ) } \left ( -2\,a{\it RootOf} \left ( \left ( {\it Arcsinh} \left ( {\it \_Z} \right ) \right ) ^{2}{a}^{2}-{{\it \_Z}}^{2}{x}^{2}-2\,{\it Arcsinh} \left ( {\it \_Z} \right ) {\it \_C1}\,a+{{\it \_C1}}^{2}-{x}^{2} \right ) \left ( a{\it Arcsinh} \left ( {\it RootOf} \left ( \left ( {\it Arcsinh} \left ( {\it \_Z} \right ) \right ) ^{2}{a}^{2}-{{\it \_Z}}^{2}{x}^{2}-2\,{\it Arcsinh} \left ( {\it \_Z} \right ) {\it \_C1}\,a+{{\it \_C1}}^{2}-{x}^{2} \right ) \right ) -{\it \_C1} \right ) \sqrt { \left ( {\it RootOf} \left ( \left ( {\it Arcsinh} \left ( {\it \_Z} \right ) \right ) ^{2}{a}^{2}-{{\it \_Z}}^{2}{x}^{2}-2\,{\it Arcsinh} \left ( {\it \_Z} \right ) {\it \_C1}\,a+{{\it \_C1}}^{2}-{x}^{2} \right ) \right ) ^{2}+1}+ \left ( \left ( {\it RootOf} \left ( \left ( {\it Arcsinh} \left ( {\it \_Z} \right ) \right ) ^{2}{a}^{2}-{{\it \_Z}}^{2}{x}^{2}-2\,{\it Arcsinh} \left ( {\it \_Z} \right ) {\it \_C1}\,a+{{\it \_C1}}^{2}-{x}^{2} \right ) \right ) ^{2}+1 \right ) \left ( \left ( {\it RootOf} \left ( \left ( {\it Arcsinh} \left ( {\it \_Z} \right ) \right ) ^{2}{a}^{2}-{{\it \_Z}}^{2}{x}^{2}-2\,{\it Arcsinh} \left ( {\it \_Z} \right ) {\it \_C1}\,a+{{\it \_C1}}^{2}-{x}^{2} \right ) \right ) ^{2}{x}^{2}+{a}^{2}+{x}^{2} \right ) \right ) } \right \}$