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 : Tuesday, March 04, 2025 at 06:56:04 PM
CAS classification : system_of_ODEs

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

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

ode:=[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)]; 
dsolve(ode);
 

\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*}

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

ode={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]}; 
ic={}; 
DSolve[{ode,ic},{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*}

✓ Sympy. Time used: 0.196 (sec). Leaf size: 90

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(-24*t - 2*x__1(t) + x__3(t) + Derivative(x__1(t), t),0),Eq(-x__1(t) + x__2(t) + Derivative(x__2(t), t),0),Eq(-3*x__1(t) + x__2(t) + x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = C_{1} + C_{2} + 2 C_{3} + t^{4} + 8 t^{3} + t^{2} \left (\frac {C_{2}}{2} + 12\right ) + t \left (2 C_{2} + C_{3}\right ), \ x^{2}{\left (t \right )} = C_{1} + \frac {C_{2} t^{2}}{2} + C_{3} + t^{4} + 4 t^{3} + t \left (C_{2} + C_{3}\right ), \ x^{3}{\left (t \right )} = 2 C_{1} + C_{2} t^{2} + 3 C_{3} + 2 t^{4} + 12 t^{3} + t \left (3 C_{2} + 2 C_{3}\right )\right ] \]