Internal problem ID [9505]
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 \relax (t )&=t x^{\prime }\relax (t )+f \left (x^{\prime }\relax (t ), y^{\prime }\relax (t )\right )\\ y \relax (t )&=y^{\prime }\relax (t ) t +g \left (x^{\prime }\relax (t ), y^{\prime }\relax (t )\right ) \end {align*}
✓ Solution by Maple
Time used: 0.187 (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))},{x(t), y(t)}, singsol=all)
\begin{align*} \int \RootOf \left (g \left (\textit {\_Z} , \frac {d}{d t}y \relax (t )\right )-y \relax (t )+t \left (\frac {d}{d t}y \relax (t )\right )\right )d t +c_{1} = \RootOf \left (g \left (\textit {\_Z} , \frac {d}{d t}y \relax (t )\right )-y \relax (t )+t \left (\frac {d}{d t}y \relax (t )\right )\right ) t +f \left (\RootOf \left (g \left (\textit {\_Z} , \frac {d}{d t}y \relax (t )\right )-y \relax (t )+t \left (\frac {d}{d t}y \relax (t )\right )\right ), \frac {d}{d t}y \relax (t )\right ) \\ \end{align*} \begin{align*} x \relax (t ) = \int \RootOf \left (g \left (\textit {\_Z} , \frac {d}{d t}y \relax (t )\right )-y \relax (t )+t \left (\frac {d}{d t}y \relax (t )\right )\right )d t +c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.003 (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*}