Internal problem ID [10249]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 9, system of higher order odes
Problem number: 1926.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x \left (t \right )&=t x^{\prime }\left (t \right )+f \left (x^{\prime }\left (t \right ), y^{\prime }\left (t \right )\right )\\ y \left (t \right )&=y^{\prime }\left (t \right ) t +g \left (x^{\prime }\left (t \right ), y^{\prime }\left (t \right )\right ) \end {align*}
✓ Solution by Maple
Time used: 0.109 (sec). Leaf size: 96
dsolve([x(t)=t*diff(x(t),t)+f(diff(x(t),t),diff(y(t),t)),y(t)=t*diff(y(t),t)+g(diff(x(t),t),diff(y(t),t))],singsol=all)
\begin{align*} \{\int \operatorname {RootOf}\left (f \left (\frac {d}{d t}x \left (t \right ), \textit {\_Z}\right )+t \left (\frac {d}{d t}x \left (t \right )\right )-x \left (t \right )\right )d t +c_{1} &= \operatorname {RootOf}\left (f \left (\frac {d}{d t}x \left (t \right ), \textit {\_Z}\right )+t \left (\frac {d}{d t}x \left (t \right )\right )-x \left (t \right )\right ) t +g \left (\frac {d}{d t}x \left (t \right ), \operatorname {RootOf}\left (f \left (\frac {d}{d t}x \left (t \right ), \textit {\_Z}\right )+t \left (\frac {d}{d t}x \left (t \right )\right )-x \left (t \right )\right )\right )\} \\ \{y \left (t \right ) &= \int \operatorname {RootOf}\left (f \left (\frac {d}{d t}x \left (t \right ), \textit {\_Z}\right )+t \left (\frac {d}{d t}x \left (t \right )\right )-x \left (t \right )\right )d t +c_{1}\} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 28
DSolve[{x[t]==t*x'[t]+f[x'[t],y'[t]],y[t]==t*y'[t]+g[x'[t],y'[t]]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to f(c_1,c_2)+c_1 t \\ y(t)\to g(c_1,c_2)+c_2 t \\ \end{align*}