9.6.29 problem problem 29

Internal problem ID [1036]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number : problem 29
Date solved : Monday, January 27, 2025 at 03:23:00 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-x_{1} \left (t \right )+x_{2} \left (t \right )+x_{3} \left (t \right )-2 x_{4} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=7 x_{1} \left (t \right )-4 x_{2} \left (t \right )-6 x_{3} \left (t \right )+11 x_{4} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=5 x_{1} \left (t \right )-x_{2} \left (t \right )+x_{3} \left (t \right )+3 x_{4} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=6 x_{1} \left (t \right )-2 x_{2} \left (t \right )-2 x_{3} \left (t \right )+6 x_{4} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.068 (sec). Leaf size: 119

dsolve([diff(x__1(t),t)=-1*x__1(t)+1*x__2(t)+1*x__3(t)-2*x__4(t),diff(x__2(t),t)=7*x__1(t)-4*x__2(t)-6*x__3(t)+11*x__4(t),diff(x__3(t),t)=5*x__1(t)-1*x__2(t)+1*x__3(t)+3*x__4(t),diff(x__4(t),t)=6*x__1(t)-2*x__2(t)-2*x__3(t)+6*x__4(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.010 (sec). Leaf size: 196

DSolve[{D[ x1[t],t]==-1*x1[t]+1*x2[t]+1*x3[t]-2*x4[t],D[ x2[t],t]==7*x1[t]-4*x2[t]-6*x3[t]+11*x4[t],D[ x3[t],t]==5*x1[t]-1*x2[t]+1*x3[t]+3*x4[t],D[ x4[t],t]==6*x1[t]-2*x2[t]-2*x3[t]+6*x4[t]},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to e^{-t} ((c_2+c_3-2 c_4) t+c_1) \\ \text {x2}(t)\to e^{-t} \left (c_1 \left (e^{3 t} (3-2 t)-3\right )-3 c_2 t-c_3 e^{3 t}-3 c_3 t+2 c_4 e^{3 t}-c_4 e^{3 t} t+6 c_4 t+c_2+c_3-2 c_4\right ) \\ \text {x3}(t)\to e^{-t} \left (c_1 \left (e^{3 t} (2 t+1)-1\right )+c_3 e^{3 t}-t \left (-c_4 \left (e^{3 t}+2\right )+c_2+c_3\right )\right ) \\ \text {x4}(t)\to e^{-t} \left (2 c_1 \left (e^{3 t}-1\right )-2 (c_2+c_3-2 c_4) t+c_4 e^{3 t}\right ) \\ \end{align*}