9.6.19 problem problem 19

Internal problem ID [1026]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number : problem 19
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 )&=x_{1} \left (t \right )-4 x_{2} \left (t \right )-2 x_{4} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{2} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=6 x_{1} \left (t \right )-12 x_{2} \left (t \right )-x_{3} \left (t \right )-6 x_{4} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=-4 x_{2} \left (t \right )-x_{4} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.024 (sec). Leaf size: 52

dsolve([diff(x__1(t),t)=1*x__1(t)-4*x__2(t)+0*x__3(t)-2*x__4(t),diff(x__2(t),t)=0*x__1(t)+1*x__2(t)+0*x__3(t)+0*x__4(t),diff(x__3(t),t)=6*x__1(t)-12*x__2(t)-1*x__3(t)-6*x__4(t),diff(x__4(t),t)=0*x__1(t)-4*x__2(t)+0*x__3(t)-1*x__4(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 114

DSolve[{D[ x1[t],t]==1*x1[t]-4*x2[t]+0*x3[t]-2*x4[t],D[ x2[t],t]==0*x1[t]+1*x2[t]+0*x3[t]+0*x4[t],D[ x3[t],t]==6*x1[t]-12*x2[t]-1*x3[t]-6*x4[t],D[ x4[t],t]==0*x1[t]-4*x2[t]+0*x3[t]-1*x4[t]},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
 

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