49.1.16 problem Example 8.3.4 from Handout chapter 8.2

Internal problem ID [9998]
Book : Selected problems from homeworks from different courses
Section : Math 2520, summer 2021. Differential Equations and Linear Algebra. Normandale college, Bloomington, Minnesota
Problem number : Example 8.3.4 from Handout chapter 8.2
Date solved : Tuesday, September 30, 2025 at 06:45:54 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=6 x \left (t \right )-7 y \left (t \right )+10\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )-2 y \left (t \right )-2 \,{\mathrm e}^{t} \end{align*}

✓ Maple. Time used: 0.186 (sec). Leaf size: 44

ode:=[diff(x(t),t) = 6*x(t)-7*y(t)+10, diff(y(t),t) = x(t)-2*y(t)-2*exp(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-t} c_2 +{\mathrm e}^{5 t} c_1 -4-\frac {7 \,{\mathrm e}^{t}}{4} \\ y \left (t \right ) &= {\mathrm e}^{-t} c_2 +\frac {{\mathrm e}^{5 t} c_1}{7}-\frac {5 \,{\mathrm e}^{t}}{4}-2 \\ \end{align*}

✓ Mathematica. Time used: 0.049 (sec). Leaf size: 90

ode={D[x[t],t]==6*x[t]-7*y[t]+10,D[y[t],t]==x[t]-2*y[t]-2*Exp[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to -\frac {7 e^t}{4}-\frac {1}{6} (c_1-7 c_2) e^{-t}+\frac {7}{6} (c_1-c_2) e^{5 t}-4\\ y(t)&\to -\frac {5 e^t}{4}-\frac {1}{6} (c_1-7 c_2) e^{-t}+\frac {1}{6} (c_1-c_2) e^{5 t}-2 \end{align*}

✓ Sympy. Time used: 0.119 (sec). Leaf size: 46

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-6*x(t) + 7*y(t) + Derivative(x(t), t) - 10,0),Eq(-x(t) + 2*y(t) + 2*exp(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = C_{1} e^{- t} + 7 C_{2} e^{5 t} - \frac {7 e^{t}}{4} - 4, \ y{\left (t \right )} = C_{1} e^{- t} + C_{2} e^{5 t} - \frac {5 e^{t}}{4} - 2\right ] \]