9.6.25 problem problem 25

Internal problem ID [1032]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number : problem 25
Date solved : Monday, January 27, 2025 at 03:22:59 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-2 x_{1} \left (t \right )+17 x_{2} \left (t \right )+4 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-x_{1} \left (t \right )+6 x_{2} \left (t \right )+x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=x_{2} \left (t \right )+2 x_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.024 (sec). Leaf size: 61

dsolve([diff(x__1(t),t)=-2*x__1(t)+17*x__2(t)+4*x__3(t),diff(x__2(t),t)=-1*x__1(t)+6*x__2(t)+1*x__3(t),diff(x__3(t),t)=0*x__1(t)+1*x__2(t)+2*x__3(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 105

DSolve[{D[ x1[t],t]==-2*x1[t]+17*x2[t]+4*x3[t],D[ x2[t],t]==-1*x1[t]+6*x2[t]+1*x3[t],D[ x3[t],t]==0*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^{2 t} \left (-\left (c_1 \left (t^2+8 t-2\right )\right )+c_2 t (4 t+34)+c_3 t (t+8)\right ) \\ \text {x2}(t)\to e^{2 t} ((-c_1+4 c_2+c_3) t+c_2) \\ \text {x3}(t)\to \frac {1}{2} e^{2 t} \left ((-c_1+4 c_2+c_3) t^2+2 c_2 t+2 c_3\right ) \\ \end{align*}