7.25.25 problem 25

Internal problem ID [645]
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 : 25
Date solved : Wednesday, February 05, 2025 at 03:51:02 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=5 x_{1} \left (t \right )+5 x_{2} \left (t \right )+2 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-6 x_{1} \left (t \right )-6 x_{2} \left (t \right )-5 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=6 x_{1} \left (t \right )+6 x_{2} \left (t \right )+5 x_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.030 (sec). Leaf size: 113

dsolve([diff(x__1(t),t)=5*x__1(t)+5*x__2(t)+2*x__3(t),diff(x__2(t),t)=-6*x__1(t)-6*x__2(t)-5*x__3(t),diff(x__3(t),t)=6*x__1(t)+6*x__2(t)+5*x__3(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= c_1 +c_2 \,{\mathrm e}^{2 t} \sin \left (3 t \right )+c_3 \,{\mathrm e}^{2 t} \cos \left (3 t \right ) \\ x_{2} \left (t \right ) &= -c_2 \,{\mathrm e}^{2 t} \sin \left (3 t \right )+c_2 \,{\mathrm e}^{2 t} \cos \left (3 t \right )-c_3 \,{\mathrm e}^{2 t} \cos \left (3 t \right )-c_3 \,{\mathrm e}^{2 t} \sin \left (3 t \right )-c_1 \\ x_{3} \left (t \right ) &= -{\mathrm e}^{2 t} \left (\cos \left (3 t \right ) c_2 -\cos \left (3 t \right ) c_3 -\sin \left (3 t \right ) c_2 -\sin \left (3 t \right ) c_3 \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.008 (sec). Leaf size: 122

DSolve[{D[x1[t],t]==5*x1[t]+5*x2[t]+2*x3[t],D[x2[t],t]==-6*x1[t]-6*x2[t]-5*x3[t],D[x3[t],t]==6*x1[t]+6*x2[t]+5*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to (c_1+c_2+c_3) e^{2 t} \cos (3 t)+(c_1+c_2) e^{2 t} \sin (3 t)-c_2-c_3 \\ \text {x2}(t)\to -c_3 e^{2 t} \cos (3 t)-(2 c_1+2 c_2+c_3) e^{2 t} \sin (3 t)+c_2+c_3 \\ \text {x3}(t)\to e^{2 t} (c_3 \cos (3 t)+(2 c_1+2 c_2+c_3) \sin (3 t)) \\ \end{align*}