69.1.137 problem 196

Internal problem ID [14290]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 196
Date solved : Tuesday, January 28, 2025 at 06:25:50 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=y \left (t \right )-x \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-x \left (t \right )-3 y \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (1\right ) = 0\\ y \left (1\right ) = 1 \end{align*}

✓ Solution by Maple

Time used: 0.042 (sec). Leaf size: 44

dsolve([diff(x(t),t) = y(t)-x(t), diff(y(t),t) = -x(t)-3*y(t), x(1) = 0, y(1) = 1], singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-2 t} \left (t \,{\mathrm e}^{2}-{\mathrm e}^{2}\right ) \\ y \left (t \right ) &= -{\mathrm e}^{-2 t} \left (t \,{\mathrm e}^{2}-2 \,{\mathrm e}^{2}\right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 31

DSolve[{D[x[t],t]==y[t]-x[t],D[y[t],t]==-x[t]-3*y[t]},{x[1]==0,y[1]==1},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to e^{2-2 t} (t-1) \\ y(t)\to -e^{2-2 t} (t-2) \\ \end{align*}