Internal problem ID [9504]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 9, system of higher order odes
Problem number: 1929.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime \prime }\left (t \right )&=-\frac {C \left (y \left (t \right )\right ) f \left (\sqrt {y^{\prime }\left (t \right )^{2}}\right ) x^{\prime }\left (t \right )}{\sqrt {x^{\prime }\left (t \right )^{2}+y^{\prime }\left (t \right )^{2}}}\\ y^{\prime \prime }\left (t \right )&=-\frac {C \left (y \left (t \right )\right ) f \left (\sqrt {y^{\prime }\left (t \right )^{2}}\right ) y^{\prime }\left (t \right )}{\sqrt {x^{\prime }\left (t \right )^{2}+y^{\prime }\left (t \right )^{2}}}-g \end {align*}
✓ Solution by Maple
Time used: 34.188 (sec). Leaf size: 1011
dsolve([diff(x(t),t,t)=-C(y(t))*f((diff(x(t),x)^2+diff(y(t),t)^2)^(1/2))/(diff(x(t),t)^2+diff(y(t),t)^2)^(1/2)*diff(x(t),t),diff(y(t),t,t)=-C(y(t))*f((diff(x(t),x)^2+diff(y(t),t)^2)^(1/2))/(diff(x(t),t)^2+diff(y(t),t)^2)^(1/2)*diff(y(t),t)-g],[x(t), y(t)], singsol=all)
\begin{align*} x \left (t \right ) = \frac {\sqrt {-\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )+\operatorname {RootOf}\left (c_{4} C \left (-\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\, t +\textit {\_C-1}\right ) f \left (\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\right ) c_{5} +c_{4} C \left (-\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\, t +\textit {\_C-1}\right ) f \left (\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\right ) t -\sqrt {\textit {\_Z}}\, c_{4} +\sqrt {\textit {\_Z}}\, c_{5} \right )}\, \left (c_{5} +t \right ) c_{4}}{c_{5}} \\ x \left (t \right ) = \frac {\sqrt {-\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )+\operatorname {RootOf}\left (C \left (-\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\, t +\textit {\_C-1}\right ) f \left (\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\right ) c_{4}^{2} t^{3}+C \left (-\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\, t +\textit {\_C-1}\right ) f \left (\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\right ) c_{4}^{2} c_{5} t +C \left (-\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\, t +\textit {\_C-1}\right ) f \left (\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\right ) c_{4} c_{5} t^{2}+C \left (-\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\, t +\textit {\_C-1}\right ) f \left (\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\right ) c_{4} c_{5}^{2}+\sqrt {\textit {\_Z}}\, c_{4}^{2} c_{5} +\sqrt {\textit {\_Z}}\, c_{5}^{2}\right )}\, \left (t^{2}+c_{5} \right ) c_{4}}{c_{4} t +c_{5}} \\ x \left (t \right ) = \frac {\sqrt {-\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right ) c_{5}^{2} c_{4}^{2}+\left (c_{4} t^{2}+c_{4} c_{5} +c_{5} t \right )^{2} C \left (-\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\, t +\textit {\_C-1}\right )^{2} f \left (\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\right )^{2} t^{2}}\, \left (c_{4} t^{2}+c_{4} c_{5} +c_{5} t \right )}{t c_{5} c_{4}^{2}} \\ x \left (t \right ) = \frac {\sqrt {-\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )+\operatorname {RootOf}\left (C \left (-\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\, t +\textit {\_C-1}\right ) f \left (\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\right ) c_{4} c_{5} t^{2}+C \left (-\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\, t +\textit {\_C-1}\right ) f \left (\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\right ) c_{4} t^{3}+C \left (-\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\, t +\textit {\_C-1}\right ) f \left (\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\right ) c_{5}^{2} t +C \left (-\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\, t +\textit {\_C-1}\right ) f \left (\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\right ) c_{5} t^{2}+\sqrt {\textit {\_Z}}\, c_{5}^{2} c_{4} -\sqrt {\textit {\_Z}}\, c_{5}^{2}\right )}\, \left (c_{4} t +c_{5} \right ) t}{\left (c_{5} +t \right ) c_{4}} \\ x \left (t \right ) = -\frac {\sqrt {-\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )+\operatorname {RootOf}\left (c_{4} C \left (-\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\, t +\textit {\_C-1}\right ) f \left (\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\right ) c_{5} +c_{4} C \left (-\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\, t +\textit {\_C-1}\right ) f \left (\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\right ) t -\sqrt {\textit {\_Z}}\, c_{4} +\sqrt {\textit {\_Z}}\, c_{5} \right )}\, \left (c_{5} +t \right ) c_{4}}{c_{5}} \\ x \left (t \right ) = -\frac {\sqrt {-\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )+\operatorname {RootOf}\left (C \left (-\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\, t +\textit {\_C-1}\right ) f \left (\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\right ) c_{4}^{2} t^{3}+C \left (-\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\, t +\textit {\_C-1}\right ) f \left (\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\right ) c_{4}^{2} c_{5} t +C \left (-\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\, t +\textit {\_C-1}\right ) f \left (\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\right ) c_{4} c_{5} t^{2}+C \left (-\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\, t +\textit {\_C-1}\right ) f \left (\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\right ) c_{4} c_{5}^{2}+\sqrt {\textit {\_Z}}\, c_{4}^{2} c_{5} +\sqrt {\textit {\_Z}}\, c_{5}^{2}\right )}\, \left (t^{2}+c_{5} \right ) c_{4}}{c_{4} t +c_{5}} \\ x \left (t \right ) = -\frac {\sqrt {-\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right ) c_{5}^{2} c_{4}^{2}+\left (c_{4} t^{2}+c_{4} c_{5} +c_{5} t \right )^{2} C \left (-\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\, t +\textit {\_C-1}\right )^{2} f \left (\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\right )^{2} t^{2}}\, \left (c_{4} t^{2}+c_{4} c_{5} +c_{5} t \right )}{t c_{5} c_{4}^{2}} \\ x \left (t \right ) = -\frac {\sqrt {-\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )+\operatorname {RootOf}\left (C \left (-\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\, t +\textit {\_C-1}\right ) f \left (\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\right ) c_{4} c_{5} t^{2}+C \left (-\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\, t +\textit {\_C-1}\right ) f \left (\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\right ) c_{4} t^{3}+C \left (-\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\, t +\textit {\_C-1}\right ) f \left (\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\right ) c_{5}^{2} t +C \left (-\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\, t +\textit {\_C-1}\right ) f \left (\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\right ) c_{5} t^{2}+\sqrt {\textit {\_Z}}\, c_{5}^{2} c_{4} -\sqrt {\textit {\_Z}}\, c_{5}^{2}\right )}\, \left (c_{4} t +c_{5} \right ) t}{\left (c_{5} +t \right ) c_{4}} \\ x \left (t \right ) = -\frac {t}{c_{2}}+c_{3} \\ \end{align*} \begin{align*} y \left (t \right ) = -\sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\, t +\textit {\_C-1} \\ y \left (t \right ) = \sqrt {\operatorname {RootOf}\left (f \left (\sqrt {\textit {\_Z}}\right )\right )}\, t +\textit {\_C-1} \\ y \left (t \right ) = \int \left (-g \left (\int \frac {1}{\frac {d}{d t}x \left (t \right )}d t \right )+\textit {\_C-1}\right ) \left (\frac {d}{d t}x \left (t \right )\right )d t +c_{1} \\ \end{align*}
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[{x''[t]==-c[y[t]]*f[(x'[t]^2+y'[t]^2)^(1/2)]/(x'[t]^2+y'[t]^2)^(1/2)*x'[t],y''[t]==-c[y[t]]*f[(x'[t]^2+y'[t]^2)^(1/2)]/(x'[t]^2+y'[t]^2)^(1/2)*y'[t]-g},{x[t],y[t]},t,IncludeSingularSolutions -> True]
Not solved