4.18.14 problem 18

Internal problem ID [1441]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 7.9, Nonhomogeneous Linear Systems. page 447
Problem number : 18
Date solved : Tuesday, September 30, 2025 at 04:34:09 AM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Maple. Time used: 0.233 (sec). Leaf size: 92

ode:=[diff(x__1(t),t) = -2*x__1(t)+x__2(t)+2*exp(-t), diff(x__2(t),t) = x__1(t)-2*x__2(t)+3*t]; 
ic:=[x__1(0) = alpha__1, x__2(0) = alpha__2]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x_{1} \left (t \right ) &= -{\mathrm e}^{-3 t} \left (\frac {2}{3}+\frac {\alpha _{2}}{2}-\frac {\alpha _{1}}{2}\right )+{\mathrm e}^{-t} \left (\frac {3}{2}+\frac {\alpha _{2}}{2}+\frac {\alpha _{1}}{2}\right )+{\mathrm e}^{-t} t +\frac {{\mathrm e}^{-t}}{2}+t -\frac {4}{3} \\ x_{2} \left (t \right ) &= {\mathrm e}^{-t} \left (\frac {3}{2}+\frac {\alpha _{2}}{2}+\frac {\alpha _{1}}{2}\right )+{\mathrm e}^{-3 t} \left (\frac {2}{3}+\frac {\alpha _{2}}{2}-\frac {\alpha _{1}}{2}\right )+{\mathrm e}^{-t} t -\frac {{\mathrm e}^{-t}}{2}+2 t -\frac {5}{3} \\ \end{align*}

✓ Mathematica. Time used: 0.04 (sec). Leaf size: 122

ode={D[ x1[t],t]==-2*x1[t]+1*x2[t]+2*Exp[-t],D[ x2[t],t]==1*x1[t]-2*x2[t]+3*t}; 
ic={x1[0]==a1,x2[0]==a2}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)&\to \frac {1}{6} e^{-3 t} \left (3 \text {a1} \left (e^{2 t}+1\right )+3 \text {a2} \left (e^{2 t}-1\right )+12 e^{2 t}-8 e^{3 t}+6 e^{2 t} t+6 e^{3 t} t-4\right )\\ \text {x2}(t)&\to \frac {1}{6} e^{-3 t} \left (3 \text {a1} \left (e^{2 t}-1\right )+3 \text {a2} \left (e^{2 t}+1\right )+6 e^{2 t} (t+1)+2 e^{3 t} (6 t-5)+4\right ) \end{align*}

✓ Sympy. Time used: 0.134 (sec). Leaf size: 56

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

\[ \left [ x^{1}{\left (t \right )} = - C_{2} e^{- 3 t} + t + t e^{- t} + \left (C_{1} + \frac {1}{2}\right ) e^{- t} - \frac {4}{3}, \ x^{2}{\left (t \right )} = C_{2} e^{- 3 t} + 2 t + t e^{- t} + \left (C_{1} - \frac {1}{2}\right ) e^{- t} - \frac {5}{3}\right ] \]