9.6.32 problem problem 32

Internal problem ID [1039]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number : problem 32
Date solved : Monday, January 27, 2025 at 03:23:02 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=11 x_{1} \left (t \right )-x_{2} \left (t \right )+26 x_{3} \left (t \right )+6 x_{4} \left (t \right )-3 x_{5} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=3 x_{2} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-9 x_{1} \left (t \right )-24 x_{3} \left (t \right )-6 x_{4} \left (t \right )+3 x_{5} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=3 x_{1} \left (t \right )+9 x_{3} \left (t \right )+5 x_{4} \left (t \right )-x_{5} \left (t \right )\\ \frac {d}{d t}x_{5} \left (t \right )&=-48 x_{1} \left (t \right )-3 x_{2} \left (t \right )-138 x_{3} \left (t \right )-30 x_{4} \left (t \right )+18 x_{5} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.074 (sec). Leaf size: 106

dsolve([diff(x__1(t),t)=11*x__1(t)-1*x__2(t)+26*x__3(t)+6*x__4(t)-3*x__5(t),diff(x__2(t),t)=0*x__1(t)+3*x__2(t)+0*x__3(t)+0*x__4(t),diff(x__3(t),t)=-9*x__1(t)+0*x__2(t)-24*x__3(t)-6*x__4(t)+3*x__5(t),diff(x__4(t),t)=3*x__1(t)+0*x__2(t)+9*x__3(t)+5*x__4(t)-1*x__5(t),diff(x__5(t),t)=-48*x__1(t)-3*x__2(t)-138*x__3(t)-30*x__4(t)+18*x__5(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= \left (-\left (c_4 +c_5 \right ) {\mathrm e}^{t}+c_1 \right ) {\mathrm e}^{2 t} \\ x_{2} \left (t \right ) &= c_5 \,{\mathrm e}^{3 t} \\ x_{3} \left (t \right ) &= {\mathrm e}^{2 t} c_3 +c_4 \,{\mathrm e}^{3 t} \\ x_{4} \left (t \right ) &= -\frac {{\mathrm e}^{2 t} c_3}{3}-\frac {c_4 \,{\mathrm e}^{3 t}}{3}+c_2 \,{\mathrm e}^{3 t} \\ x_{5} \left (t \right ) &= 8 \,{\mathrm e}^{2 t} c_3 +\frac {16 c_4 \,{\mathrm e}^{3 t}}{3}+2 c_2 \,{\mathrm e}^{3 t}-3 c_5 \,{\mathrm e}^{3 t}+3 \,{\mathrm e}^{2 t} c_1 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.011 (sec). Leaf size: 211

DSolve[{D[ x1[t],t]==11*x1[t]-1*x2[t]+26*x3[t]+6*x4[t]-3*x5[t],D[ x2[t],t]==0*x1[t]+3*x2[t],D[ x3[t],t]==-9*x1[t]+0*x2[t]-24*x3[t]-6*x4[t]+3*x5[t],D[ x4[t],t]==3*x1[t]+0*x2[t]+9*x3[t]+5*x4[t]-1*x5[t],D[ x5[t],t]==-48*x1[t]-3*x2[t]-138*x3[t]-30*x4[t]+18*x5[t]},{x1[t],x2[t],x3[t],x4[t],x5[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to e^{2 t} \left (c_1 \left (9 e^t-8\right )-(c_2-26 c_3-6 c_4+3 c_5) \left (e^t-1\right )\right ) \\ \text {x2}(t)\to c_2 e^{3 t} \\ \text {x3}(t)\to -e^{2 t} \left (9 c_1 \left (e^t-1\right )+c_3 \left (26 e^t-27\right )+3 (2 c_4-c_5) \left (e^t-1\right )\right ) \\ \text {x4}(t)\to e^{2 t} \left (3 c_1 \left (e^t-1\right )+9 c_3 \left (e^t-1\right )+3 c_4 e^t-c_5 e^t-2 c_4+c_5\right ) \\ \text {x5}(t)\to -e^{2 t} \left (48 c_1 \left (e^t-1\right )+3 c_2 \left (e^t-1\right )+138 c_3 e^t+30 c_4 e^t-16 c_5 e^t-138 c_3-30 c_4+15 c_5\right ) \\ \end{align*}