75.29.1 problem 802

Internal problem ID [17255]
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 22. Integration of homogeneous linear systems with constant coefficients. Eulers method. Exercises page 230
Problem number : 802
Date solved : Tuesday, January 28, 2025 at 09:58:28 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=8 y \left (t \right )-x \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )+y \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.051 (sec). Leaf size: 35

dsolve([diff(x(t),t)=8*y(t)-x(t),diff(y(t),t)=x(t)+y(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-3 t} c_{1} +c_{2} {\mathrm e}^{3 t} \\ y \left (t \right ) &= -\frac {{\mathrm e}^{-3 t} c_{1}}{4}+\frac {c_{2} {\mathrm e}^{3 t}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 72

DSolve[{D[x[t],t]==8*y[t]-x[t],D[y[t],t]==x[t]+y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{3} e^{-3 t} \left (c_1 \left (e^{6 t}+2\right )+4 c_2 \left (e^{6 t}-1\right )\right ) \\ y(t)\to \frac {1}{6} e^{-3 t} \left (c_1 \left (e^{6 t}-1\right )+2 c_2 \left (2 e^{6 t}+1\right )\right ) \\ \end{align*}