9.6.33 problem problem 33

Internal problem ID [1040]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number : problem 33
Date solved : Wednesday, February 05, 2025 at 02:33:15 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.418 (sec). Leaf size: 139

dsolve([diff(x__1(t),t)=3*x__1(t)-4*x__2(t)+1*x__3(t)+0*x__4(t),diff(x__2(t),t)=4*x__1(t)+3*x__2(t)+0*x__3(t)+1*x__4(t),diff(x__3(t),t)=0*x__1(t)+0*x__2(t)+3*x__3(t)-4*x__4(t),diff(x__4(t),t)=0*x__1(t)+0*x__2(t)+4*x__3(t)+3*x__4(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= \frac {{\mathrm e}^{3 t} \left (4 \cos \left (4 t \right ) c_4 t +4 \sin \left (4 t \right ) c_3 t +4 c_1 \cos \left (4 t \right )+4 c_2 \sin \left (4 t \right )-\sin \left (4 t \right ) c_4 \right )}{4} \\ x_{2} \left (t \right ) &= -\frac {{\mathrm e}^{3 t} \left (4 \cos \left (4 t \right ) c_3 t -4 \sin \left (4 t \right ) c_4 t +4 c_2 \cos \left (4 t \right )-c_4 \cos \left (4 t \right )-4 c_1 \sin \left (4 t \right )\right )}{4} \\ x_{3} \left (t \right ) &= {\mathrm e}^{3 t} \left (c_4 \cos \left (4 t \right )+c_3 \sin \left (4 t \right )\right ) \\ x_{4} \left (t \right ) &= -{\mathrm e}^{3 t} \left (\cos \left (4 t \right ) c_3 -\sin \left (4 t \right ) c_4 \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.092 (sec). Leaf size: 120

DSolve[{D[ x1[t],t]==3*x1[t]-4*x2[t]+1*x3[t]+0*x4[t],D[ x2[t],t]==4*x1[t]+3*x2[t]+0*x3[t]+1*x4[t],D[ x3[t],t]==0*x1[t]+0*x2[t]+3*x3[t]-4*x4[t],D[ x4[t],t]==0*x1[t]+0*x2[t]+4*x3[t]+3*x4[t]},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to e^{3 t} ((c_3 t+c_1) \cos (4 t)-(c_4 t+c_2) \sin (4 t)) \\ \text {x2}(t)\to e^{3 t} ((c_4 t+c_2) \cos (4 t)+(c_3 t+c_1) \sin (4 t)) \\ \text {x3}(t)\to e^{3 t} (c_3 \cos (4 t)-c_4 \sin (4 t)) \\ \text {x4}(t)\to e^{3 t} (c_4 \cos (4 t)+c_3 \sin (4 t)) \\ \end{align*}