52.9.13 problem 13

Internal problem ID [8391]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 8 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS. EXERCISES 8.1. Page 332
Problem number : 13
Date solved : Monday, January 27, 2025 at 03:57:25 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=-x \left (t \right )+\frac {y}{4}\\ y^{\prime }&=x \left (t \right )-y \end{align*}

✓ Solution by Maple

Time used: 0.022 (sec). Leaf size: 35

dsolve([diff(x(t),t)=-x(t)+1/4*y(t),diff(y(t),t)=x(t)-y(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 66

DSolve[{D[x[t],t]==-x[t]+1/4*y[t],D[y[t],t]==x[t]-y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

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