28.4.52 problem 7.52

Internal problem ID [4584]
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 : 7.52
Date solved : Monday, January 27, 2025 at 09:25:41 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.370 (sec). Leaf size: 98

dsolve([diff(x__1(t),t)=2*x__1(t)-x__2(t)+2*x__3(t),diff(x__2(t),t)=x__1(t)+2*x__3(t),diff(x__3(t),t)=-2*x__1(t)+x__2(t)-x__3(t)+4*sin(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= \cos \left (t \right ) c_{1} -4 \cos \left (t \right ) t +c_3 \sin \left (t \right )-4 \cos \left (t \right )+4 \sin \left (t \right ) \\ x_{2} \left (t \right ) &= \cos \left (t \right ) c_{1} -4 \cos \left (t \right ) t +c_3 \sin \left (t \right )+c_{2} {\mathrm e}^{t}-4 \cos \left (t \right )+4 \sin \left (t \right ) \\ x_{3} \left (t \right ) &= -\frac {\cos \left (t \right ) c_{1}}{2}+\frac {\cos \left (t \right ) c_3}{2}+2 \cos \left (t \right ) t -\frac {c_{1} \sin \left (t \right )}{2}-\frac {c_3 \sin \left (t \right )}{2}+2 \sin \left (t \right ) t +\frac {c_{2} {\mathrm e}^{t}}{2}+2 \cos \left (t \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.035 (sec). Leaf size: 125

DSolve[{D[x1[t],t]==2*x1[t]-x2[t]+2*x3[t],D[x2[t],t]==x1[t]+2*x3[t],D[x3[t],t]==-2*x1[t]+x2[t]-x3[t]+4*Sin[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

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