87.20.12 problem 13

Internal problem ID [23697]
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 : 13
Date solved : Thursday, October 02, 2025 at 09:44:22 PM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x_{1} \left (0\right )&=-{\frac {5}{18}} \\ x_{2} \left (0\right )&={\frac {47}{9}} \\ \end{align*}

✓ Maple. Time used: 0.081 (sec). Leaf size: 37

ode:=[diff(x__1(t),t) = 4*x__1(t)-x__2(t)+3*exp(2*t), diff(x__2(t),t) = 2*x__1(t)+x__2(t)+2*t]; 
ic:=[x__1(0) = -5/18, x__2(0) = 47/9]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x_{1} \left (t \right ) &= -\frac {5}{18}-3 \,{\mathrm e}^{2 t} t -\frac {t}{3} \\ x_{2} \left (t \right ) &= -\frac {7}{9}+6 \,{\mathrm e}^{2 t}-6 \,{\mathrm e}^{2 t} t -\frac {4 t}{3} \\ \end{align*}

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 45

ode={D[x1[t],t]==4*x1[t]-x2[t]+3*Exp[2*t],D[x2[t],t]==2*x1[t]+x2[t]+2*t}; 
ic={x1[0]==-5/18,x2[0]==47/9}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)&\to -\frac {1}{3} \left (9 e^{2 t}+1\right ) t-\frac {5}{18}\\ \text {x2}(t)&\to -6 e^{2 t} (t-1)-\frac {4 t}{3}-\frac {7}{9} \end{align*}

✓ Sympy. Time used: 0.216 (sec). Leaf size: 44

from sympy import * 
t = symbols("t") 
x1 = Function("x1") 
x2 = Function("x2") 
ode=[Eq(-4*x1(t) + x2(t) - 3*exp(2*t) + Derivative(x1(t), t),0),Eq(-2*t - 2*x1(t) - x2(t) + Derivative(x2(t), t),0)] 
ics = {x1(0): -5/18, x2(0): 47/9} 
dsolve(ode,func=[x1(t),x2(t)],ics=ics)
 

\[ \left [ x_{1}{\left (t \right )} = - 3 t e^{2 t} - \frac {t}{3} - \frac {5}{18}, \ x_{2}{\left (t \right )} = - 6 t e^{2 t} - \frac {4 t}{3} + 6 e^{2 t} - \frac {7}{9}\right ] \]