42.5.6 problem Problem 5.7

Internal problem ID [8857]
Book : THEORY OF DIFFERENTIAL EQUATIONS IN ENGINEERING AND MECHANICS. K.T. CHAU, CRC Press. Boca Raton, FL. 2018
Section : Chapter 5. Systems of First Order Differential Equations. Section 5.11 Problems. Page 360
Problem number : Problem 5.7
Date solved : Tuesday, September 30, 2025 at 05:57:57 PM
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*}

✓ Maple. Time used: 0.184 (sec). Leaf size: 64

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]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= -{\mathrm e}^{-3 t} c_2 +{\mathrm e}^{-t} c_1 +{\mathrm e}^{-t} t +\frac {{\mathrm e}^{-t}}{2}+t -\frac {4}{3} \\ x_{2} \left (t \right ) &= {\mathrm e}^{-3 t} c_2 +{\mathrm e}^{-t} c_1 -\frac {{\mathrm e}^{-t}}{2}+{\mathrm e}^{-t} t -\frac {5}{3}+2 t \\ \end{align*}

✓ Mathematica. Time used: 0.027 (sec). Leaf size: 256

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

\begin{align*} \text {x1}(t)&\to \frac {1}{2} e^{-3 t} \left (\left (e^{2 t}+1\right ) \int _1^t\left (\frac {3}{2} e^{K[1]} K[1]-\frac {3}{2} e^{3 K[1]} K[1]+e^{2 K[1]}+1\right )dK[1]+\left (e^{2 t}-1\right ) \int _1^t\left (\frac {3}{2} e^{K[2]} K[2]+\frac {3}{2} e^{3 K[2]} K[2]-e^{2 K[2]}+1\right )dK[2]+c_1 e^{2 t}+c_2 e^{2 t}+c_1-c_2\right )\\ \text {x2}(t)&\to \frac {1}{2} e^{-3 t} \left (\left (e^{2 t}-1\right ) \int _1^t\left (\frac {3}{2} e^{K[1]} K[1]-\frac {3}{2} e^{3 K[1]} K[1]+e^{2 K[1]}+1\right )dK[1]+\left (e^{2 t}+1\right ) \int _1^t\left (\frac {3}{2} e^{K[2]} K[2]+\frac {3}{2} e^{3 K[2]} K[2]-e^{2 K[2]}+1\right )dK[2]+c_1 e^{2 t}+c_2 e^{2 t}-c_1+c_2\right ) \end{align*}

✓ Sympy. Time used: 0.128 (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)
 

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