14.23.3 problem 3

Internal problem ID [2742]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Section 3.9, Systems of differential equations. Complex roots. Page 344
Problem number : 3
Date solved : Monday, January 27, 2025 at 06:12:35 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.153 (sec). Leaf size: 69

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

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

✓ Solution by Mathematica

Time used: 0.009 (sec). Leaf size: 91

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

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