28.4.51 problem 7.51

Internal problem ID [4583]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 7. Systems of linear differential equations. Problems at page 351
Problem number : 7.51
Date solved : Monday, January 27, 2025 at 09:25:39 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.160 (sec). Leaf size: 74

dsolve([diff(x__1(t),t)=x__1(t)-x__2(t)-x__3(t)+4*exp(t),diff(x__2(t),t)=x__1(t)+x__2(t),diff(x__3(t),t)=3*x__1(t)+x__3(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.071 (sec). Leaf size: 158

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

\begin{align*} \text {x1}(t)\to c_1 \cos \left (\sqrt {3} t\right )-\frac {(c_1+c_2+c_3) \sin \left (\sqrt {3} t\right )}{\sqrt {3}} \\ \text {x2}(t)\to \frac {1}{12} \left (3 (4+3 c_2-c_3) e^t+3 (c_2+c_3) \cos \left (\sqrt {3} t\right )+\sqrt {3} (4 c_1+c_2+c_3) \sin \left (\sqrt {3} t\right )\right ) \\ \text {x3}(t)\to \frac {1}{4} \left ((12-3 c_2+c_3) e^t+3 (c_2+c_3) \cos \left (\sqrt {3} t\right )+\sqrt {3} (4 c_1+c_2+c_3) \sin \left (\sqrt {3} t\right )\right ) \\ \end{align*}