87.20.4 problem 4

Internal problem ID [23689]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 3. Linear Systems. Exercise at page 161
Problem number : 4
Date solved : Thursday, October 02, 2025 at 09:44:17 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.048 (sec). Leaf size: 38

ode:=[diff(x__1(t),t) = 3*x__1(t)-2*x__2(t)+1, diff(x__2(t),t) = x__1(t)+t]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.204 (sec). Leaf size: 75

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

\begin{align*} \text {x1}(t)&\to -t-(c_1-2 c_2) e^t+2 (c_1-c_2) e^{2 t}-\frac {3}{2}\\ \text {x2}(t)&\to -\frac {3 t}{2}-(c_1-2 c_2) e^t+(c_1-c_2) e^{2 t}-\frac {5}{4} \end{align*}

✓ Sympy. Time used: 0.120 (sec). Leaf size: 42

from sympy import * 
t = symbols("t") 
x1 = Function("x1") 
x2 = Function("x2") 
ode=[Eq(-3*x1(t) + 2*x2(t) + Derivative(x1(t), t) - 1,0),Eq(-t - x1(t) + Derivative(x2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x1(t),x2(t)],ics=ics)
 

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