75.30.1 problem 810

Internal problem ID [17192]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 3 (Systems of differential equations). Section 23. Methods of integrating nonhomogeneous linear systems with constant coefficients. Exercises page 234
Problem number : 810
Date solved : Monday, March 31, 2025 at 03:44:14 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 )&=-{\mathrm e}^{2 t}\\ \frac {d}{d t}y \left (t \right )+3 x \left (t \right )-2 y \left (t \right )&=6 \,{\mathrm e}^{2 t} \end{align*}

✓ Maple. Time used: 0.173 (sec). Leaf size: 42

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

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-t} c_2 +{\mathrm e}^{t} c_1 +2 \,{\mathrm e}^{2 t} \\ y \left (t \right ) &= {\mathrm e}^{-t} c_2 +3 \,{\mathrm e}^{t} c_1 +9 \,{\mathrm e}^{2 t} \\ \end{align*}

✓ Mathematica. Time used: 0.148 (sec). Leaf size: 85

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

\begin{align*} x(t)\to \frac {1}{2} e^{-t} \left (4 e^{3 t}+(c_2-c_1) e^{2 t}+3 c_1-c_2\right ) \\ y(t)\to \frac {1}{2} e^{-t} \left (18 e^{3 t}-3 (c_1-c_2) e^{2 t}+3 c_1-c_2\right ) \\ \end{align*}

✓ Sympy. Time used: 0.138 (sec). Leaf size: 39

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

\[ \left [ x{\left (t \right )} = C_{1} e^{- t} + \frac {C_{2} e^{t}}{3} + 2 e^{2 t}, \ y{\left (t \right )} = C_{1} e^{- t} + C_{2} e^{t} + 9 e^{2 t}\right ] \]