28.4.55 problem 7.55

Internal problem ID [4587]
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 : 7.55
Date solved : Monday, January 27, 2025 at 09:25:46 AM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=2 x_{1} \left (t \right )-x_{3} \left (t \right )+24 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_{2} \left (t \right )-x_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.072 (sec). Leaf size: 103

dsolve([diff(x__1(t),t)=2*x__1(t)-x__3(t)+24*t,diff(x__2(t),t)=x__1(t)-x__2(t),diff(x__3(t),t)=3*x__1(t)-x__2(t)-x__3(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= t^{4}+8 t^{3}+\frac {1}{2} c_{1} t^{2}+c_{2} t +c_3 \\ x_{2} \left (t \right ) &= -12 t^{2}+c_{1} +24 t +4 t^{3}-c_{1} t -c_{2} -24+t^{4}+\frac {1}{2} c_{1} t^{2}+c_{2} t +c_3 \\ x_{3} \left (t \right ) &= 2 t^{4}+c_{1} t^{2}+12 t^{3}-c_{1} t +2 c_{2} t -24 t^{2}+2 c_3 -c_{2} +24 t \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 127

DSolve[{D[x1[t],t]==2*x1[t]-x3[t]+24*t,D[x2[t],t]==x1[t]-x2[t],D[x3[t],t]==3*x1[t]-x2[t]-x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to t^4+8 t^3+\frac {1}{2} (24+c_1+c_2-c_3) t^2+(2 c_1-c_3) t+c_1 \\ \text {x2}(t)\to t^4+4 t^3+\frac {1}{2} (c_1+c_2-c_3) t^2+(c_1-c_2) t+c_2 \\ \text {x3}(t)\to 2 t^4+12 t^3+(c_1+c_2-c_3) t^2+(3 c_1-c_2-c_3) t+c_3 \\ \end{align*}