9.6.20 problem problem 20

Internal problem ID [1027]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number : problem 20
Date solved : Monday, January 27, 2025 at 03:22:58 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.050 (sec). Leaf size: 77

dsolve([diff(x__1(t),t)=2*x__1(t)+1*x__2(t)+0*x__3(t)+1*x__4(t),diff(x__2(t),t)=0*x__1(t)+2*x__2(t)+1*x__3(t)+0*x__4(t),diff(x__3(t),t)=0*x__1(t)+0*x__2(t)+2*x__3(t)+1*x__4(t),diff(x__4(t),t)=0*x__1(t)+0*x__2(t)+0*x__3(t)+2*x__4(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 96

DSolve[{D[ x1[t],t]==2*x1[t]+1*x2[t]+0*x3[t]+1*x4[t],D[ x2[t],t]==0*x1[t]+2*x2[t]+1*x3[t]+0*x4[t],D[ x3[t],t]==0*x1[t]+0*x2[t]+2*x3[t]+1*x4[t],D[ x4[t],t]==0*x1[t]+0*x2[t]+0*x3[t]+2*x4[t]},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
 

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