14.26.13 problem 13

Internal problem ID [2786]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 3. Systems of differential equations. Section 3.13 (Solving systems by Laplace transform). Page 370
Problem number : 13
Date solved : Sunday, March 30, 2025 at 12:20:29 AM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 0\\ x_{2} \left (0\right ) = 0\\ x_{3} \left (0\right ) = 0 \end{align*}

✓ Maple. Time used: 0.500 (sec). Leaf size: 3512

ode:=[diff(x__1(t),t) = -x__1(t)-x__2(t)+2*x__3(t)+exp(t), diff(x__2(t),t) = x__1(t)+x__2(t)+x__3(t), diff(x__3(t),t) = 2*x__1(t)+x__2(t)+3*x__3(t)]; 
ic:=x__1(0) = 0x__2(0) = 0x__3(0) = 0; 
dsolve([ode,ic]);
 

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Mathematica. Time used: 0.065 (sec). Leaf size: 2327

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

Too large to display

✗ Sympy

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

Timed Out