70.28.1 problem 2

Internal problem ID [19154]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 6. Systems of First Order Linear Equations. Section 6.6 (Nonhomogeneous Linear Systems). Problems at page 436
Problem number : 2
Date solved : Thursday, October 02, 2025 at 03:38:43 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 )+{\mathrm e}^{t}\\ \frac {d}{d t}x_{2} \left (t \right )&=3 x_{1} \left (t \right )-2 x_{2} \left (t \right )+t \end{align*}

✓ Maple. Time used: 0.163 (sec). Leaf size: 53

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

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

✓ Mathematica. Time used: 0.35 (sec). Leaf size: 269

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

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

✓ Sympy. Time used: 0.108 (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) - exp(t) + Derivative(x__1(t), t),0),Eq(-t - 3*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)
 

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