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 = 25.6773 (sec), leaf count = 110

DSolve[-x^2 + y[x]^2 - 2*x*y[x]*Derivative[1][y][x] + a*y[x]*Sqrt[1 + Derivative[1][y][x]^2] == 0,y[x],x]
 

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

✓ Maple : cpu = 1.163 (sec), leaf count = 1058

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

\[\int _{\textit {\_b}}^{x}\frac {2 \textit {\_a}^{3}-2 \textit {\_a} y \left (x \right )^{2}+\sqrt {a^{2} \left (y \left (x \right )^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) y \left (x \right )^{2}+\textit {\_a}^{4}\right )}}{-2 a^{2} \textit {\_a} y \left (x \right )^{2}+2 \textit {\_a}^{5}+4 \textit {\_a}^{3} y \left (x \right )^{2}+2 y \left (x \right )^{4} \textit {\_a} +\textit {\_a}^{2} \sqrt {a^{2} \left (y \left (x \right )^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) y \left (x \right )^{2}+\textit {\_a}^{4}\right )}-y \left (x \right )^{2} \sqrt {a^{2} \left (y \left (x \right )^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) y \left (x \right )^{2}+\textit {\_a}^{4}\right )}}d \textit {\_a} +\int _{}^{y \left (x \right )}\left (-\frac {\left (a^{2}-4 x^{2}\right ) \textit {\_f}}{-2 a^{2} x \,\textit {\_f}^{2}+2 x^{5}+4 x^{3} \textit {\_f}^{2}+2 x \,\textit {\_f}^{4}+x^{2} \sqrt {a^{2} \left (\textit {\_f}^{4}-\left (a^{2}-2 x^{2}\right ) \textit {\_f}^{2}+x^{4}\right )}-\textit {\_f}^{2} \sqrt {a^{2} \left (\textit {\_f}^{4}-\left (a^{2}-2 x^{2}\right ) \textit {\_f}^{2}+x^{4}\right )}}-\left (\int _{\textit {\_b}}^{x}\frac {2 \textit {\_f} \left (\frac {a^{2} \textit {\_a} \,\textit {\_f}^{2} \left (a -2 \textit {\_a} \right ) \left (2 \textit {\_a} +a \right ) \left (-2 \textit {\_a}^{2}-2 \textit {\_f}^{2}+a^{2}\right )}{\sqrt {a^{2} \left (\textit {\_f}^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) \textit {\_f}^{2}+\textit {\_a}^{4}\right )}}+2 \textit {\_a} \left (-2 \textit {\_a}^{2}-2 \textit {\_f}^{2}+a^{2}\right ) \sqrt {a^{2} \left (\textit {\_f}^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) \textit {\_f}^{2}+\textit {\_a}^{4}\right )}-12 \textit {\_a}^{6}+\left (-8 \textit {\_f}^{2}+5 a^{2}\right ) \textit {\_a}^{4}+\left (4 \textit {\_f}^{4}+2 \textit {\_f}^{2} a^{2}\right ) \textit {\_a}^{2}-\textit {\_f}^{2} a^{4}+\textit {\_f}^{4} a^{2}\right )}{\left (2 a^{2} \textit {\_a} \,\textit {\_f}^{2}-2 \textit {\_a}^{5}-4 \textit {\_a}^{3} \textit {\_f}^{2}-2 \textit {\_f}^{4} \textit {\_a} -\textit {\_a}^{2} \sqrt {a^{2} \left (\textit {\_f}^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) \textit {\_f}^{2}+\textit {\_a}^{4}\right )}+\textit {\_f}^{2} \sqrt {a^{2} \left (\textit {\_f}^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) \textit {\_f}^{2}+\textit {\_a}^{4}\right )}\right )^{2}}d \textit {\_a} \right )\right )d \textit {\_f} +c_{1} = 0\]