9.4.18 problem problem 18

Internal problem ID [982]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.3, The eigenvalue method for linear systems. Page 395
Problem number : problem 18
Date solved : Monday, January 27, 2025 at 03:22:43 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{1} \left (t \right )+2 x_{2} \left (t \right )+2 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=2 x_{1} \left (t \right )+7 x_{2} \left (t \right )+x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=2 x_{1} \left (t \right )+x_{2} \left (t \right )+7 x_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 53

dsolve([diff(x__1(t),t)=x__1(t)+2*x__2(t)+2*x__3(t),diff(x__2(t),t)=2*x__1(t)+7*x__2(t)+x__3(t),diff(x__3(t),t)=2*x__1(t)+x__2(t)+7*x__3(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= c_2 +c_3 \,{\mathrm e}^{9 t} \\ x_{2} \left (t \right ) &= 2 c_3 \,{\mathrm e}^{9 t}+c_1 \,{\mathrm e}^{6 t}-\frac {c_2}{4} \\ x_{3} \left (t \right ) &= 2 c_3 \,{\mathrm e}^{9 t}-c_1 \,{\mathrm e}^{6 t}-\frac {c_2}{4} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 148

DSolve[{D[ x1[t],t]==x1[t]+2*x2[t]+2*x3[t],D[ x2[t],t]==2*x1[t]+7*x2[t]+x3[t],D[ x3[t],t]==2*x1[t]+x2[t]+7*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{9} \left (c_1 \left (e^{9 t}+8\right )+2 (c_2+c_3) \left (e^{9 t}-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{18} \left (4 c_1 \left (e^{9 t}-1\right )+c_2 \left (9 e^{6 t}+8 e^{9 t}+1\right )+c_3 \left (-9 e^{6 t}+8 e^{9 t}+1\right )\right ) \\ \text {x3}(t)\to \frac {1}{18} \left (4 c_1 \left (e^{9 t}-1\right )+c_2 \left (-9 e^{6 t}+8 e^{9 t}+1\right )+c_3 \left (9 e^{6 t}+8 e^{9 t}+1\right )\right ) \\ \end{align*}