7.25.8 problem 8

Internal problem ID [628]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 5. Linear systems of differential equations. Section 5.4 (The eigenvalue method for homogeneous systems). Problems at page 378
Problem number : 8
Date solved : Wednesday, February 05, 2025 at 03:50:55 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: 0.012 (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*}