28.4.15 problem 7.15

Internal problem ID [4547]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 7. Systems of linear differential equations. Problems at page 351
Problem number : 7.15
Date solved : Sunday, March 30, 2025 at 03:25:59 AM
CAS classification : system_of_ODEs

\begin{align*} 3 \frac {d}{d t}x \left (t \right )+2 x \left (t \right )+\frac {d}{d t}y \left (t \right )-6 y \left (t \right )&=5 \,{\mathrm e}^{t}\\ 4 \frac {d}{d t}x \left (t \right )+2 x \left (t \right )+\frac {d}{d t}y \left (t \right )-8 y \left (t \right )&=5 \,{\mathrm e}^{t}+2 t -3 \end{align*}

✓ Maple. Time used: 0.226 (sec). Leaf size: 71

ode:=[3*diff(x(t),t)+2*x(t)+diff(y(t),t)-6*y(t) = 5*exp(t), 4*diff(x(t),t)+2*x(t)+diff(y(t),t)-8*y(t) = 5*exp(t)+2*t-3]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.296 (sec). Leaf size: 54

ode={3*D[x[t],t]+2*x[t]+D[y[t],t]-6*y[t]==5*Exp[t],4*D[x[t],t]+2*x[t]+D[y[t],t]-8*y[t]==5*Exp[t]+2*t-3}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to -3 t+2 e^t+c_1 \cos (2 t)+c_2 \sin (2 t)+5 \\ y(t)\to -t+e^t+c_2 \cos (2 t)-c_1 \sin (2 t) \\ \end{align*}

✓ Sympy. Time used: 0.295 (sec). Leaf size: 129

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(2*x(t) - 6*y(t) - 5*exp(t) + 3*Derivative(x(t), t) + Derivative(y(t), t),0),Eq(-2*t + 2*x(t) - 8*y(t) - 5*exp(t) + 4*Derivative(x(t), t) + Derivative(y(t), t) + 3,0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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