80.7.13 problem B 5

Internal problem ID [21332]
Book : A Textbook on Ordinary Differential Equations by Shair Ahmad and Antonio Ambrosetti. Second edition. ISBN 978-3-319-16407-6. Springer 2015
Section : Chapter 7. System of first order equations. Excercise 7.6 at page 162
Problem number : B 5
Date solved : Thursday, October 02, 2025 at 07:28:30 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }&=2 x+6 y \left (t \right )+{\mathrm e}^{t}\\ y^{\prime }\left (t \right )&=x+3 y \left (t \right )-{\mathrm e}^{t} \end{align*}

✓ Maple. Time used: 0.067 (sec). Leaf size: 35

ode:=[diff(x(t),t) = 2*x(t)+6*y(t)+exp(t), diff(y(t),t) = x(t)+3*y(t)-exp(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= \frac {{\mathrm e}^{5 t} c_1}{5}+2 \,{\mathrm e}^{t}+c_2 \\ y \left (t \right ) &= \frac {{\mathrm e}^{5 t} c_1}{10}-\frac {{\mathrm e}^{t}}{2}-\frac {c_2}{3} \\ \end{align*}

✓ Mathematica. Time used: 0.061 (sec). Leaf size: 73

ode={D[x[t],t]==2*x[t]+6*y[t]+Exp[t],D[y[t],t]==x[t]+3*y[t]-Exp[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to 2 e^t+\frac {2}{5} (c_1+3 c_2) e^{5 t}+\frac {3}{5} (c_1-2 c_2)\\ y(t)&\to \frac {1}{10} \left (-5 e^t+2 (c_1+3 c_2) e^{5 t}-2 c_1+4 c_2\right ) \end{align*}

✓ Sympy. Time used: 0.078 (sec). Leaf size: 34

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-2*x(t) - 6*y(t) - exp(t) + Derivative(x(t), t),0),Eq(-x(t) - 3*y(t) + exp(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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