9.6.22 problem problem 22

Internal problem ID [1029]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number : problem 22
Date solved : Monday, January 27, 2025 at 03:22:59 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{1} \left (t \right )+3 x_{2} \left (t \right )+7 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-x_{2} \left (t \right )-4 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=x_{2} \left (t \right )+3 x_{3} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=-6 x_{2} \left (t \right )-14 x_{3} \left (t \right )+x_{4} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.046 (sec). Leaf size: 77

dsolve([diff(x__1(t),t)=1*x__1(t)+3*x__2(t)+7*x__3(t)+0*x__4(t),diff(x__2(t),t)=0*x__1(t)-1*x__2(t)-4*x__3(t)+0*x__4(t),diff(x__3(t),t)=0*x__1(t)+1*x__2(t)+3*x__3(t)+0*x__4(t),diff(x__4(t),t)=0*x__1(t)-6*x__2(t)-14*x__3(t)+1*x__4(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= \frac {\left (-c_4 \,t^{2}-2 c_3 t -7 c_4 t +4 c_2 \right ) {\mathrm e}^{t}}{4} \\ x_{2} \left (t \right ) &= {\mathrm e}^{t} \left (c_4 t +c_3 \right ) \\ x_{3} \left (t \right ) &= -\frac {{\mathrm e}^{t} \left (2 c_4 t +2 c_3 +c_4 \right )}{4} \\ x_{4} \left (t \right ) &= \frac {\left (c_4 \,t^{2}+2 c_3 t +7 c_4 t +2 c_1 \right ) {\mathrm e}^{t}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 99

DSolve[{D[ x1[t],t]==1*x1[t]+3*x2[t]+7*x3[t]+0*x4[t],D[ x2[t],t]==0*x1[t]-1*x2[t]-4*x3[t]+0*x4[t],D[ x3[t],t]==0*x1[t]+1*x2[t]+3*x3[t]+0*x4[t],D[ x4[t],t]==0*x1[t]-6*x2[t]-14*x3[t]+1*x4[t]},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{2} e^t (c_2 t (t+6)+2 c_3 t (t+7)+2 c_1) \\ \text {x2}(t)\to -e^t (c_2 (2 t-1)+4 c_3 t) \\ \text {x3}(t)\to e^t ((c_2+2 c_3) t+c_3) \\ \text {x4}(t)\to e^t (c_2 (-t) (t+6)-2 c_3 t (t+7)+c_4) \\ \end{align*}