7.25.13 problem 13

Internal problem ID [633]
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 : 13
Date solved : Wednesday, February 05, 2025 at 03:50:59 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=5 x_{1} \left (t \right )-9 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=2 x_{1} \left (t \right )-x_{2} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 59

dsolve([diff(x__1(t),t)=5*x__1(t)-9*x__2(t),diff(x__2(t),t)=2*x__1(t)-x__2(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{2 t} \left (\sin \left (3 t \right ) c_1 +\cos \left (3 t \right ) c_2 \right ) \\ x_{2} \left (t \right ) &= -\frac {{\mathrm e}^{2 t} \left (-\sin \left (3 t \right ) c_1 -\sin \left (3 t \right ) c_2 +\cos \left (3 t \right ) c_1 -\cos \left (3 t \right ) c_2 \right )}{3} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 66

DSolve[{D[x1[t],t]==5*x1[t]-9*x2[t],D[x2[t],t]==2*x1[t]-x2[t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to e^{2 t} (c_1 \cos (3 t)+(c_1-3 c_2) \sin (3 t)) \\ \text {x2}(t)\to \frac {1}{3} e^{2 t} (3 c_2 \cos (3 t)+(2 c_1-3 c_2) \sin (3 t)) \\ \end{align*}