#### 2.560   ODE No. 560

$a y(x) \sqrt {y'(x)^2+1}-x^2-2 x y(x) y'(x)+y(x)^2=0$ Mathematica : cpu = 13.7502 (sec), leaf count = 110

$\left \{\left \{y(x)\to -\frac {\sqrt {a^2 c_1{}^2 \left (-x^2\right )-4 a^2 c_1 x-4 a^2+4 x^2}}{\sqrt {-4+a^2 c_1{}^2}}\right \},\left \{y(x)\to \frac {\sqrt {a^2 c_1{}^2 \left (-x^2\right )-4 a^2 c_1 x-4 a^2+4 x^2}}{\sqrt {-4+a^2 c_1{}^2}}\right \}\right \}$ Maple : cpu = 1.16 (sec), leaf count = 1120

$\left \{ \int _{{\it \_b}}^{x}\!{ \left ( 2\,{{\it \_a}}^{3}-2\,{\it \_a}\, \left ( y \left ( x \right ) \right ) ^{2}+\sqrt {{a}^{2} \left ( {{\it \_a}}^{4}+2\,{{\it \_a}}^{2} \left ( y \left ( x \right ) \right ) ^{2}-{a}^{2} \left ( y \left ( x \right ) \right ) ^{2}+ \left ( y \left ( x \right ) \right ) ^{4} \right ) } \right ) \left ( -2\,{a}^{2}{\it \_a}\, \left ( y \left ( x \right ) \right ) ^{2}+2\,{{\it \_a}}^{5}+4\,{{\it \_a}}^{3} \left ( y \left ( x \right ) \right ) ^{2}+2\, \left ( y \left ( x \right ) \right ) ^{4}{\it \_a}+{{\it \_a}}^{2}\sqrt {{a}^{2} \left ( {{\it \_a}}^{4}+2\,{{\it \_a}}^{2} \left ( y \left ( x \right ) \right ) ^{2}-{a}^{2} \left ( y \left ( x \right ) \right ) ^{2}+ \left ( y \left ( x \right ) \right ) ^{4} \right ) }- \left ( y \left ( x \right ) \right ) ^{2}\sqrt {{a}^{2} \left ( {{\it \_a}}^{4}+2\,{{\it \_a}}^{2} \left ( y \left ( x \right ) \right ) ^{2}-{a}^{2} \left ( y \left ( x \right ) \right ) ^{2}+ \left ( y \left ( x \right ) \right ) ^{4} \right ) } \right ) ^{-1}}\,{\rm d}{\it \_a}+\int ^{y \left ( x \right ) }\!-{ \left ( {a}^{2}-4\,{x}^{2} \right ) {\it \_f} \left ( -2\,{a}^{2}x{{\it \_f}}^{2}+2\,{x}^{5}+4\,{x}^{3}{{\it \_f}}^{2}+2\,x{{\it \_f}}^{4}+{x}^{2}\sqrt {{{\it \_f}}^{4}{a}^{2}-{a}^{2} \left ( {a}^{2}-2\,{x}^{2} \right ) {{\it \_f}}^{2}+{a}^{2}{x}^{4}}-{{\it \_f}}^{2}\sqrt {{{\it \_f}}^{4}{a}^{2}-{a}^{2} \left ( {a}^{2}-2\,{x}^{2} \right ) {{\it \_f}}^{2}+{a}^{2}{x}^{4}} \right ) ^{-1}}-\int _{{\it \_b}}^{x}\!2\,{\frac {{\it \_f}}{ \left ( 2\,{a}^{2}{\it \_a}\,{{\it \_f}}^{2}-2\,{{\it \_a}}^{5}-4\,{{\it \_a}}^{3}{{\it \_f}}^{2}-2\,{{\it \_f}}^{4}{\it \_a}-{{\it \_a}}^{2}\sqrt {{a}^{2} \left ( {{\it \_a}}^{4}+2\,{{\it \_a}}^{2}{{\it \_f}}^{2}+{{\it \_f}}^{4}-{{\it \_f}}^{2}{a}^{2} \right ) }+{{\it \_f}}^{2}\sqrt {{a}^{2} \left ( {{\it \_a}}^{4}+2\,{{\it \_a}}^{2}{{\it \_f}}^{2}+{{\it \_f}}^{4}-{{\it \_f}}^{2}{a}^{2} \right ) } \right ) ^{2}} \left ( {\frac {{a}^{2}{\it \_a}\,{{\it \_f}}^{2} \left ( a-2\,{\it \_a} \right ) \left ( 2\,{\it \_a}+a \right ) \left ( -2\,{{\it \_a}}^{2}-2\,{{\it \_f}}^{2}+{a}^{2} \right ) }{\sqrt {{a}^{2} \left ( {{\it \_a}}^{4}+2\,{{\it \_a}}^{2}{{\it \_f}}^{2}+{{\it \_f}}^{4}-{{\it \_f}}^{2}{a}^{2} \right ) }}}+2\,{\it \_a}\, \left ( -2\,{{\it \_a}}^{2}-2\,{{\it \_f}}^{2}+{a}^{2} \right ) \sqrt {{a}^{2} \left ( {{\it \_a}}^{4}+2\,{{\it \_a}}^{2}{{\it \_f}}^{2}+{{\it \_f}}^{4}-{{\it \_f}}^{2}{a}^{2} \right ) }-12\,{{\it \_a}}^{6}+ \left ( -8\,{{\it \_f}}^{2}+5\,{a}^{2} \right ) {{\it \_a}}^{4}+ \left ( 4\,{{\it \_f}}^{4}+2\,{{\it \_f}}^{2}{a}^{2} \right ) {{\it \_a}}^{2}-{{\it \_f}}^{2}{a}^{4}+{{\it \_f}}^{4}{a}^{2} \right ) }\,{\rm d}{\it \_a}{d{\it \_f}}+{\it \_C1}=0,\int _{{\it \_b}}^{x}\!{ \left ( 2\,{\it \_a}\, \left ( y \left ( x \right ) \right ) ^{2}-2\,{{\it \_a}}^{3}+\sqrt {{a}^{2} \left ( {{\it \_a}}^{4}+2\,{{\it \_a}}^{2} \left ( y \left ( x \right ) \right ) ^{2}-{a}^{2} \left ( y \left ( x \right ) \right ) ^{2}+ \left ( y \left ( x \right ) \right ) ^{4} \right ) } \right ) \left ( 2\,{a}^{2}{\it \_a}\, \left ( y \left ( x \right ) \right ) ^{2}-2\,{{\it \_a}}^{5}-4\,{{\it \_a}}^{3} \left ( y \left ( x \right ) \right ) ^{2}-2\, \left ( y \left ( x \right ) \right ) ^{4}{\it \_a}+{{\it \_a}}^{2}\sqrt {{a}^{2} \left ( {{\it \_a}}^{4}+2\,{{\it \_a}}^{2} \left ( y \left ( x \right ) \right ) ^{2}-{a}^{2} \left ( y \left ( x \right ) \right ) ^{2}+ \left ( y \left ( x \right ) \right ) ^{4} \right ) }- \left ( y \left ( x \right ) \right ) ^{2}\sqrt {{a}^{2} \left ( {{\it \_a}}^{4}+2\,{{\it \_a}}^{2} \left ( y \left ( x \right ) \right ) ^{2}-{a}^{2} \left ( y \left ( x \right ) \right ) ^{2}+ \left ( y \left ( x \right ) \right ) ^{4} \right ) } \right ) ^{-1}}\,{\rm d}{\it \_a}+\int ^{y \left ( x \right ) }\!{ \left ( {a}^{2}-4\,{x}^{2} \right ) {\it \_f} \left ( 2\,{a}^{2}x{{\it \_f}}^{2}-2\,{x}^{5}-4\,{x}^{3}{{\it \_f}}^{2}-2\,x{{\it \_f}}^{4}+{x}^{2}\sqrt {{{\it \_f}}^{4}{a}^{2}-{a}^{2} \left ( {a}^{2}-2\,{x}^{2} \right ) {{\it \_f}}^{2}+{a}^{2}{x}^{4}}-{{\it \_f}}^{2}\sqrt {{{\it \_f}}^{4}{a}^{2}-{a}^{2} \left ( {a}^{2}-2\,{x}^{2} \right ) {{\it \_f}}^{2}+{a}^{2}{x}^{4}} \right ) ^{-1}}-\int _{{\it \_b}}^{x}\!-2\,{\frac {{\it \_f}}{ \left ( 2\,{a}^{2}{\it \_a}\,{{\it \_f}}^{2}-2\,{{\it \_a}}^{5}-4\,{{\it \_a}}^{3}{{\it \_f}}^{2}-2\,{{\it \_f}}^{4}{\it \_a}+{{\it \_a}}^{2}\sqrt {{a}^{2} \left ( {{\it \_a}}^{4}+2\,{{\it \_a}}^{2}{{\it \_f}}^{2}+{{\it \_f}}^{4}-{{\it \_f}}^{2}{a}^{2} \right ) }-{{\it \_f}}^{2}\sqrt {{a}^{2} \left ( {{\it \_a}}^{4}+2\,{{\it \_a}}^{2}{{\it \_f}}^{2}+{{\it \_f}}^{4}-{{\it \_f}}^{2}{a}^{2} \right ) } \right ) ^{2}} \left ( {\frac {{a}^{2}{\it \_a}\,{{\it \_f}}^{2} \left ( a-2\,{\it \_a} \right ) \left ( 2\,{\it \_a}+a \right ) \left ( -2\,{{\it \_a}}^{2}-2\,{{\it \_f}}^{2}+{a}^{2} \right ) }{\sqrt {{a}^{2} \left ( {{\it \_a}}^{4}+2\,{{\it \_a}}^{2}{{\it \_f}}^{2}+{{\it \_f}}^{4}-{{\it \_f}}^{2}{a}^{2} \right ) }}}+2\,{\it \_a}\, \left ( -2\,{{\it \_a}}^{2}-2\,{{\it \_f}}^{2}+{a}^{2} \right ) \sqrt {{a}^{2} \left ( {{\it \_a}}^{4}+2\,{{\it \_a}}^{2}{{\it \_f}}^{2}+{{\it \_f}}^{4}-{{\it \_f}}^{2}{a}^{2} \right ) }+12\,{{\it \_a}}^{6}+ \left ( 8\,{{\it \_f}}^{2}-5\,{a}^{2} \right ) {{\it \_a}}^{4}+ \left ( -4\,{{\it \_f}}^{4}-2\,{{\it \_f}}^{2}{a}^{2} \right ) {{\it \_a}}^{2}+{{\it \_f}}^{2}{a}^{4}-{{\it \_f}}^{4}{a}^{2} \right ) }\,{\rm d}{\it \_a}{d{\it \_f}}+{\it \_C1}=0 \right \}$