14.24.8 problem 6

Internal problem ID [2755]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Section 3.10, Systems of differential equations. Equal roots. Page 352
Problem number : 6
Date solved : Monday, January 27, 2025 at 06:12:47 AM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=-4 x_{1} \left (t \right )-4 x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=10 x_{1} \left (t \right )+9 x_{2} \left (t \right )+x_{3} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=-4 x_{1} \left (t \right )-3 x_{2} \left (t \right )+x_{3} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 2\\ x_{2} \left (0\right ) = 1\\ x_{3} \left (0\right ) = -1 \end{align*}

✓ Solution by Maple

Time used: 0.042 (sec). Leaf size: 57

dsolve([diff(x__1(t),t) = -4*x__1(t)-4*x__2(t), diff(x__2(t),t) = 10*x__1(t)+9*x__2(t)+x__3(t), diff(x__3(t),t) = -4*x__1(t)-3*x__2(t)+x__3(t), x__1(0) = 2, x__2(0) = 1, x__3(0) = -1], singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{2 t} \left (-4 t^{2}-16 t +2\right ) \\ x_{2} \left (t \right ) &= -\frac {{\mathrm e}^{2 t} \left (-24 t^{2}-104 t -4\right )}{4} \\ x_{3} \left (t \right ) &= \frac {{\mathrm e}^{2 t} \left (-8 t^{2}-40 t -4\right )}{4} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 61

DSolve[{D[ x1[t],t]==-4*x1[t]-4*x2[t]+0*x3[t],D[ x2[t],t]==10*x1[t]+9*x2[t]+1*x3[t],D[ x3[t],t]==-4*x1[t]-3*x2[t]+1*x3[t]},{x1[0]==2,x2[0]==1,x3[0]==-1},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to -2 e^{2 t} \left (2 t^2+8 t-1\right ) \\ \text {x2}(t)\to e^{2 t} \left (6 t^2+26 t+1\right ) \\ \text {x3}(t)\to -e^{2 t} \left (2 t^2+10 t+1\right ) \\ \end{align*}