7.25.12 problem 12

Internal problem ID [632]
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 : 12
Date solved : Wednesday, February 05, 2025 at 03:50:58 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 )+3 x_{2} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.017 (sec). Leaf size: 58

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

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

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 67

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

\begin{align*} \text {x1}(t)\to \frac {1}{2} e^{2 t} (2 c_1 \cos (2 t)-(c_1+5 c_2) \sin (2 t)) \\ \text {x2}(t)\to \frac {1}{2} e^{2 t} (2 c_2 \cos (2 t)+(c_1+c_2) \sin (2 t)) \\ \end{align*}