60.7.237 problem 1828 (book 6.237)

Internal problem ID [11826]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1828 (book 6.237)
Date solved : Monday, January 27, 2025 at 11:43:08 PM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} a^{2} {y^{\prime \prime }}^{2}-2 a x y^{\prime \prime }+y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.125 (sec). Leaf size: 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), singsol=all)
 

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

✗ Solution by Mathematica

Time used: 0.000 (sec). Leaf size: 0

DSolve[D[y[x],x] - 2*a*x*D[y[x],{x,2}] + a^2*D[y[x],{x,2}]^2 == 0,y[x],x,IncludeSingularSolutions -> True]
 

Not solved