9.6.23 problem problem 23

Internal problem ID [1030]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number : problem 23
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 )&=39 x_{1} \left (t \right )+8 x_{2} \left (t \right )-16 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-36 x_{1} \left (t \right )-5 x_{2} \left (t \right )+16 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=72 x_{1} \left (t \right )+16 x_{2} \left (t \right )-29 x_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.029 (sec). Leaf size: 66

dsolve([diff(x__1(t),t)=39*x__1(t)+8*x__2(t)-16*x__3(t),diff(x__2(t),t)=-36*x__1(t)-5*x__2(t)+16*x__3(t),diff(x__3(t),t)=72*x__1(t)+16*x__2(t)-29*x__3(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= c_2 \,{\mathrm e}^{3 t}+c_3 \,{\mathrm e}^{-t} \\ x_{2} \left (t \right ) &= -c_2 \,{\mathrm e}^{3 t}-c_3 \,{\mathrm e}^{-t}+c_1 \,{\mathrm e}^{3 t} \\ x_{3} \left (t \right ) &= \frac {7 c_2 \,{\mathrm e}^{3 t}}{4}+2 c_3 \,{\mathrm e}^{-t}+\frac {c_1 \,{\mathrm e}^{3 t}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 127

DSolve[{D[ x1[t],t]==39*x1[t]+8*x2[t]-16*x3[t],D[ x2[t],t]==-36*x1[t]-5*x2[t]+16*x3[t],D[ x3[t],t]==72*x1[t]+16*x2[t]-29*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to e^{-t} \left (c_1 \left (10 e^{4 t}-9\right )+2 (c_2-2 c_3) \left (e^{4 t}-1\right )\right ) \\ \text {x2}(t)\to e^{-t} \left (-9 c_1 \left (e^{4 t}-1\right )-c_2 \left (e^{4 t}-2\right )+4 c_3 \left (e^{4 t}-1\right )\right ) \\ \text {x3}(t)\to e^{-t} \left (18 c_1 \left (e^{4 t}-1\right )+4 c_2 \left (e^{4 t}-1\right )+c_3 \left (8-7 e^{4 t}\right )\right ) \\ \end{align*}