75.27.2 problem 777

Internal problem ID [17238]
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
Date solved : Tuesday, January 28, 2025 at 09:58:19 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=t +y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )-t \end{align*}

✓ Solution by Maple

Time used: 0.034 (sec). Leaf size: 36

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}+c_{1} {\mathrm e}^{t}+t -1 \\ y \left (t \right ) &= -c_{2} {\mathrm e}^{-t}+c_{1} {\mathrm e}^{t}+1-t \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 78

DSolve[{D[x[t],t]==y[t]+t,D[y[t],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*}