7.25.9 problem 9

Internal problem ID [629]
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 : 9
Date solved : Wednesday, February 05, 2025 at 03:50:56 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=2 x_{1} \left (t \right )-5 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=4 x_{1} \left (t \right )-2 x_{2} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 2\\ x_{2} \left (0\right ) = 3 \end{align*}

✓ Solution by Maple

Time used: 0.018 (sec). Leaf size: 33

dsolve([diff(x__1(t),t) = 2*x__1(t)-5*x__2(t), diff(x__2(t),t) = 4*x__1(t)-2*x__2(t), x__1(0) = 2, x__2(0) = 3], singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= -\frac {11 \sin \left (4 t \right )}{4}+2 \cos \left (4 t \right ) \\ x_{2} \left (t \right ) &= 3 \cos \left (4 t \right )+\frac {\sin \left (4 t \right )}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 38

DSolve[{D[x1[t],t]==2*x1[t]-5*x2[t],D[x2[t],t]==4*x1[t]-2*x2[t]},{x1[0]==2,x2[0]==3},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to 2 \cos (4 t)-\frac {11}{4} \sin (4 t) \\ \text {x2}(t)\to \frac {1}{2} (\sin (4 t)+6 \cos (4 t)) \\ \end{align*}