9.4.8 problem problem 8

Internal problem ID [972]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.3, The eigenvalue method for linear systems. Page 395
Problem number : problem 8
Date solved : Wednesday, February 05, 2025 at 04:51:20 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{1} \left (t \right )-5 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )-x_{2} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 49

dsolve([diff(x__1(t),t)=x__1(t)-5*x__2(t),diff(x__2(t),t)=x__1(t)-x__2(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= c_1 \sin \left (2 t \right )+c_2 \cos \left (2 t \right ) \\ x_{2} \left (t \right ) &= -\frac {2 c_1 \cos \left (2 t \right )}{5}+\frac {2 c_2 \sin \left (2 t \right )}{5}+\frac {c_1 \sin \left (2 t \right )}{5}+\frac {c_2 \cos \left (2 t \right )}{5} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 48

DSolve[{D[ x1[t],t]==x1[t]-5*x2[t],D[ x2[t],t]==x1[t]-x2[t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to c_1 \cos (2 t)+(c_1-5 c_2) \sin (t) \cos (t) \\ \text {x2}(t)\to c_2 \cos (2 t)+(c_1-c_2) \sin (t) \cos (t) \\ \end{align*}