76.28.1 problem 2

Internal problem ID [17789]
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, March 13, 2025 at 10:49:19 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 )+{\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: 0.093 (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 {c_{2} {\mathrm e}^{-t}}{3}+c_{1} {\mathrm e}^{t}+\frac {3 \,{\mathrm e}^{t} t}{2}-\frac {{\mathrm e}^{t}}{4}+t \\ x_{2} \left (t \right ) &= c_{2} {\mathrm e}^{-t}+c_{1} {\mathrm e}^{t}+\frac {3 \,{\mathrm e}^{t} t}{2}-\frac {3 \,{\mathrm e}^{t}}{4}+2 t -1 \\ \end{align*}

✓ Mathematica. Time used: 0.213 (sec). Leaf size: 97

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}{4} e^{-t} \left (4 e^t t+e^{2 t} (6 t-1+6 c_1-2 c_2)-2 c_1+2 c_2\right ) \\ \text {x2}(t)\to \frac {1}{4} e^{-t} \left (e^t (8 t-4)+e^{2 t} (6 t-3+6 c_1-2 c_2)-6 c_1+6 c_2\right ) \\ \end{align*}

✓ Sympy. Time used: 0.178 (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)
 

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