49.1.15 problem HW 5 problem 7

Internal problem ID [9997]
Book : Selected problems from homeworks from different courses
Section : Math 2520, summer 2021. Differential Equations and Linear Algebra. Normandale college, Bloomington, Minnesota
Problem number : HW 5 problem 7
Date solved : Tuesday, September 30, 2025 at 06:45:53 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.161 (sec). Leaf size: 44

ode:=[diff(x(t),t) = 2*x(t)-y(t), diff(y(t),t) = -x(t)+2*y(t)+4*exp(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 74

ode={D[x[t],t]==2*x[t]-y[t],D[y[t],t]==-x[t]+2*y[t]+4*Exp[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.084 (sec). Leaf size: 44

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-2*x(t) + y(t) + Derivative(x(t), t),0),Eq(x(t) - 2*y(t) - 4*exp(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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