2.1827   ODE No. 1827

\[ a^2 y''(x)^2-2 a x y''(x)+y'(x)=0 \]

✗ Mathematica : cpu = 0.798585 (sec), leaf count = 0

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

, could not solve

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

✓ Maple : cpu = 4.732 (sec), leaf count = 1364

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

\[y \left (x \right ) = \int \operatorname {RootOf}\left (8 \textit {\_Z}^{-2 a +1} a^{3} \sqrt {x^{2}-\textit {\_Z}}\, \left (-x +\sqrt {x^{2}-\textit {\_Z}}\right )^{-2 a} \left (x +\sqrt {x^{2}-\textit {\_Z}}\right )^{2 a} \left (4 \textit {\_Z} \,a^{2}-4 a \,x^{2}+x^{2}\right )^{-2 a} \left (2 a \sqrt {x^{2}-\textit {\_Z}}-2 a x +x \right )^{2 a} \left (2 a \sqrt {x^{2}-\textit {\_Z}}+2 a x -x \right )^{-2 a}-8 \textit {\_Z}^{-2 a} \left (2 a \sqrt {x^{2}-\textit {\_Z}}+2 a x -x \right )^{-2 a} \left (2 a \sqrt {x^{2}-\textit {\_Z}}-2 a x +x \right )^{2 a} \sqrt {x^{2}-\textit {\_Z}}\, \left (-x +\sqrt {x^{2}-\textit {\_Z}}\right )^{-2 a} \left (4 \textit {\_Z} \,a^{2}-4 a \,x^{2}+x^{2}\right )^{-2 a} \left (x +\sqrt {x^{2}-\textit {\_Z}}\right )^{2 a} a^{2} x^{2}+8 x \,\textit {\_Z}^{-2 a +1} a^{3} \left (-x +\sqrt {x^{2}-\textit {\_Z}}\right )^{-2 a} \left (x +\sqrt {x^{2}-\textit {\_Z}}\right )^{2 a} \left (4 \textit {\_Z} \,a^{2}-4 a \,x^{2}+x^{2}\right )^{-2 a} \left (2 a \sqrt {x^{2}-\textit {\_Z}}-2 a x +x \right )^{2 a} \left (2 a \sqrt {x^{2}-\textit {\_Z}}+2 a x -x \right )^{-2 a}-8 \textit {\_Z}^{-2 a} \left (2 a \sqrt {x^{2}-\textit {\_Z}}+2 a x -x \right )^{-2 a} \left (2 a \sqrt {x^{2}-\textit {\_Z}}-2 a x +x \right )^{2 a} \left (-x +\sqrt {x^{2}-\textit {\_Z}}\right )^{-2 a} \left (4 \textit {\_Z} \,a^{2}-4 a \,x^{2}+x^{2}\right )^{-2 a} \left (x +\sqrt {x^{2}-\textit {\_Z}}\right )^{2 a} a^{2} x^{3}+2 \textit {\_Z}^{-2 a} \left (2 a \sqrt {x^{2}-\textit {\_Z}}+2 a x -x \right )^{-2 a} \left (2 a \sqrt {x^{2}-\textit {\_Z}}-2 a x +x \right )^{2 a} \sqrt {x^{2}-\textit {\_Z}}\, \left (-x +\sqrt {x^{2}-\textit {\_Z}}\right )^{-2 a} \left (4 \textit {\_Z} \,a^{2}-4 a \,x^{2}+x^{2}\right )^{-2 a} \left (x +\sqrt {x^{2}-\textit {\_Z}}\right )^{2 a} a \,x^{2}-4 x \,\textit {\_Z}^{-2 a +1} a^{2} \left (-x +\sqrt {x^{2}-\textit {\_Z}}\right )^{-2 a} \left (x +\sqrt {x^{2}-\textit {\_Z}}\right )^{2 a} \left (4 \textit {\_Z} \,a^{2}-4 a \,x^{2}+x^{2}\right )^{-2 a} \left (2 a \sqrt {x^{2}-\textit {\_Z}}-2 a x +x \right )^{2 a} \left (2 a \sqrt {x^{2}-\textit {\_Z}}+2 a x -x \right )^{-2 a}+6 \textit {\_Z}^{-2 a} \left (2 a \sqrt {x^{2}-\textit {\_Z}}+2 a x -x \right )^{-2 a} \left (2 a \sqrt {x^{2}-\textit {\_Z}}-2 a x +x \right )^{2 a} \left (-x +\sqrt {x^{2}-\textit {\_Z}}\right )^{-2 a} \left (4 \textit {\_Z} \,a^{2}-4 a \,x^{2}+x^{2}\right )^{-2 a} \left (x +\sqrt {x^{2}-\textit {\_Z}}\right )^{2 a} a \,x^{3}-\textit {\_Z}^{-2 a} \left (2 a \sqrt {x^{2}-\textit {\_Z}}+2 a x -x \right )^{-2 a} \left (2 a \sqrt {x^{2}-\textit {\_Z}}-2 a x +x \right )^{2 a} \left (-x +\sqrt {x^{2}-\textit {\_Z}}\right )^{-2 a} \left (4 \textit {\_Z} \,a^{2}-4 a \,x^{2}+x^{2}\right )^{-2 a} \left (x +\sqrt {x^{2}-\textit {\_Z}}\right )^{2 a} x^{3}-2 \sqrt {x^{2}-\textit {\_Z}}\, c_{1} a +2 c_{1} a x -x c_{1} \right )d x +c_{2}\]