8.25.7 problem 5

Internal problem ID [2764]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Section 3.12, Systems of differential equations. The nonhomogeneous equation. variation of parameters. Page 366
Problem number : 5
Date solved : Tuesday, September 30, 2025 at 05:50:47 AM
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 )+{\mathrm e}^{2 t}\\ \frac {d}{d t}x_{2} \left (t \right )&=2 x_{2} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=x_{2} \left (t \right )+3 x_{3} \left (t \right )+{\mathrm e}^{2 t} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.231 (sec). Leaf size: 44

ode:=[diff(x__1(t),t) = 2*x__1(t)+x__3(t)+exp(2*t), diff(x__2(t),t) = 2*x__2(t), diff(x__3(t),t) = x__2(t)+3*x__3(t)+exp(2*t)]; 
ic:=[x__1(0) = 1, x__2(0) = 1, x__3(0) = 1]; 
dsolve([ode,op(ic)]);
 

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

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 45

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

\begin{align*} \text {x1}(t)&\to e^{2 t} \left (-t+3 e^t-2\right )\\ \text {x2}(t)&\to e^{2 t}\\ \text {x3}(t)&\to e^{2 t} \left (3 e^t-2\right ) \end{align*}

✓ Sympy. Time used: 0.102 (sec). Leaf size: 53

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

\[ \left [ x^{1}{\left (t \right )} = C_{1} t e^{2 t} + C_{3} e^{3 t} + \left (C_{2} - 1\right ) e^{2 t}, \ x^{2}{\left (t \right )} = - C_{1} e^{2 t}, \ x^{3}{\left (t \right )} = C_{3} e^{3 t} + \left (C_{1} - 1\right ) e^{2 t}\right ] \]