Internal problem ID [15508]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Chapter 3 (Systems of differential equations). Section 20. The method of elimination.
Exercises page 212
Problem number: 777.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=y \left (t \right )+t\\ y^{\prime }\left (t \right )&=x \left (t \right )-t \end {align*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 37
dsolve([diff(x(t),t)=y(t)+t,diff(y(t),t)=x(t)-t],singsol=all)
\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{t}+{\mathrm e}^{-t} c_{1} +t -1 \\ y \left (t \right ) &= c_{2} {\mathrm e}^{t}-{\mathrm e}^{-t} c_{1} +1-t \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.014 (sec). Leaf size: 78
DSolve[{x'[t]==y[t]+t,y'[t]==x[t]-t},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {1}{2} e^{-t} \left (2 e^t (t-1)+(c_1+c_2) e^{2 t}+c_1-c_2\right ) \\ y(t)\to \frac {1}{2} e^{-t} \left (-2 e^t (t-1)+(c_1+c_2) e^{2 t}-c_1+c_2\right ) \\ \end{align*}