9.6.31 problem problem 31

Internal problem ID [1038]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number : problem 31
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 )&=35 x_{1} \left (t \right )-12 x_{2} \left (t \right )+4 x_{3} \left (t \right )+30 x_{4} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=22 x_{1} \left (t \right )-8 x_{2} \left (t \right )+3 x_{3} \left (t \right )+19 x_{4} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-10 x_{1} \left (t \right )+3 x_{2} \left (t \right )-9 x_{4} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=-27 x_{1} \left (t \right )+9 x_{2} \left (t \right )-3 x_{3} \left (t \right )-23 x_{4} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.054 (sec). Leaf size: 116

dsolve([diff(x__1(t),t)=35*x__1(t)-12*x__2(t)+4*x__3(t)+30*x__4(t),diff(x__2(t),t)=22*x__1(t)-8*x__2(t)+3*x__3(t)+19*x__4(t),diff(x__3(t),t)=-10*x__1(t)+3*x__2(t)+0*x__3(t)-9*x__4(t),diff(x__4(t),t)=-27*x__1(t)+9*x__2(t)-3*x__3(t)-23*x__4(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= \frac {{\mathrm e}^{t} \left (-6 c_4 \,t^{2}-6 c_3 t -4 c_4 t +3 c_1 -6 c_2 -2 c_3 \right )}{3} \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{t} \left (-3 c_4 \,t^{2}-3 c_3 t -16 c_4 t +3 c_1 -3 c_2 -8 c_3 +6 c_4 \right )}{9} \\ x_{3} \left (t \right ) &= {\mathrm e}^{t} \left (c_4 \,t^{2}+c_3 t +c_2 \right ) \\ x_{4} \left (t \right ) &= -\frac {{\mathrm e}^{t} \left (-18 c_4 \,t^{2}-18 c_3 t -6 c_4 t +9 c_1 -18 c_2 -3 c_3 -2 c_4 \right )}{9} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 207

DSolve[{D[ x1[t],t]==35*x1[t]-12*x2[t]+4*x3[t]+30*x4[t],D[ x2[t],t]==22*x1[t]-8*x2[t]+3*x3[t]+19*x4[t],D[ x3[t],t]==-10*x1[t]+3*x2[t]+0*x3[t]-9*x4[t],D[ x4[t],t]==-27*x1[t]+9*x2[t]-3*x3[t]-23*x4[t]},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to e^t \left (c_1 \left (21 t^2+34 t+1\right )-3 c_2 t (3 t+4)+c_3 t (3 t+4)+6 c_4 t (3 t+5)\right ) \\ \text {x2}(t)\to \frac {1}{2} e^t \left ((7 c_1-3 c_2+c_3+6 c_4) t^2+2 (22 c_1-9 c_2+3 c_3+19 c_4) t+2 c_2\right ) \\ \text {x3}(t)\to \frac {1}{2} e^t \left (-3 (7 c_1-3 c_2+c_3+6 c_4) t^2-2 (10 c_1-3 c_2+c_3+9 c_4) t+2 c_3\right ) \\ \text {x4}(t)\to e^t \left (-3 (7 c_1-3 c_2+c_3+6 c_4) t^2-3 (9 c_1-3 c_2+c_3+8 c_4) t+c_4\right ) \\ \end{align*}