9.6.27 problem problem 27

Internal problem ID [1034]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number : problem 27
Date solved : Monday, January 27, 2025 at 03:23:00 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-3 x_{1} \left (t \right )+5 x_{2} \left (t \right )-5 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=3 x_{1} \left (t \right )-x_{2} \left (t \right )+3 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=8 x_{1} \left (t \right )-8 x_{2} \left (t \right )+10 x_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.036 (sec). Leaf size: 57

dsolve([diff(x__1(t),t)=-3*x__1(t)+5*x__2(t)-5*x__3(t),diff(x__2(t),t)=3*x__1(t)-1*x__2(t)+3*x__3(t),diff(x__3(t),t)=8*x__1(t)-8*x__2(t)+10*x__3(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{2 t} \left (c_3 t +c_2 \right ) \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{2 t} \left (-3 c_3 t +5 c_1 -3 c_2 \right )}{5} \\ x_{3} \left (t \right ) &= \frac {{\mathrm e}^{2 t} \left (-8 c_3 t +5 c_1 -8 c_2 -c_3 \right )}{5} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.029 (sec). Leaf size: 174

DSolve[{D[ x1[t],t]==-3*x1[t]+5*x2[t]-5*x3[t],D[ x2[t],t]==4*x1[t]-1*x2[t]+4*x3[t],D[ x3[t],t]==8*x1[t]-8*x2[t]+10*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{3} e^{2 t} \left (-5 (c_1+c_3) \cos \left (\sqrt {3} t\right )-5 \sqrt {3} (c_1-c_2+c_3) \sin \left (\sqrt {3} t\right )+8 c_1+5 c_3\right ) \\ \text {x2}(t)\to \frac {1}{3} e^{2 t} \left (3 c_2 \cos \left (\sqrt {3} t\right )+\sqrt {3} (4 c_1-3 c_2+4 c_3) \sin \left (\sqrt {3} t\right )\right ) \\ \text {x3}(t)\to \frac {1}{3} e^{2 t} \left (8 (c_1+c_3) \cos \left (\sqrt {3} t\right )+8 \sqrt {3} (c_1-c_2+c_3) \sin \left (\sqrt {3} t\right )-8 c_1-5 c_3\right ) \\ \end{align*}