9.6.14 problem problem 14

Internal problem ID [1021]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number : problem 14
Date solved : Monday, January 27, 2025 at 03:22:57 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.019 (sec). Leaf size: 71

dsolve([diff(x__1(t),t)=0*x__1(t)+0*x__2(t)+1*x__3(t),diff(x__2(t),t)=-5*x__1(t)-1*x__2(t)-5*x__3(t),diff(x__3(t),t)=4*x__1(t)+1*x__2(t)-2*x__3(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 119

DSolve[{D[ x1[t],t]==0*x1[t]+0*x2[t]+1*x3[t],D[ x2[t],t]==-5*x1[t]-1*x2[t]-5*x3[t],D[ x3[t],t]==4*x1[t]+1*x2[t]-2*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{2} e^{-t} \left (c_1 \left (5 t^2+2 t+2\right )+t (c_2 t+2 c_3)\right ) \\ \text {x2}(t)\to \frac {1}{2} e^{-t} \left (-5 (5 c_1+c_2) t^2-10 (c_1+c_3) t+2 c_2\right ) \\ \text {x3}(t)\to \frac {1}{2} e^{-t} \left (-\left ((5 c_1+c_2) t^2\right )+2 (4 c_1+c_2-c_3) t+2 c_3\right ) \\ \end{align*}