28.4.24 problem 7.24

Internal problem ID [4556]
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 : 7.24
Date solved : Sunday, March 30, 2025 at 03:26:12 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d^{2}}{d t^{2}}x \left (t \right )+\frac {d}{d t}x \left (t \right )+\frac {d}{d t}y \left (t \right )-2 y \left (t \right )&=40 \,{\mathrm e}^{3 t}\\ \frac {d}{d t}x \left (t \right )+x \left (t \right )-\frac {d}{d t}y \left (t \right )&=36 \,{\mathrm e}^{t} \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 1\\ y \left (0\right ) = 3\\ D\left (x \right )\left (0\right ) = 1 \end{align*}

✓ Maple. Time used: 0.228 (sec). Leaf size: 57

ode:=[diff(diff(x(t),t),t)+diff(x(t),t)+diff(y(t),t)-2*y(t) = 40*exp(3*t), diff(x(t),t)+x(t)-diff(y(t),t) = 36*exp(t)]; 
ic:=x(0) = 1y(0) = 3D(x)(0) = 1; 
dsolve([ode,ic]);
 

\begin{align*} x \left (t \right ) &= -33 \,{\mathrm e}^{-t}+3 \,{\mathrm e}^{3 t}+22 \,{\mathrm e}^{-2 t}+9 \,{\mathrm e}^{t}-6 \,{\mathrm e}^{t} t \\ y \left (t \right ) &= 4 \,{\mathrm e}^{3 t}+11 \,{\mathrm e}^{-2 t}-12 \,{\mathrm e}^{t}-12 \,{\mathrm e}^{t} t \\ \end{align*}

✓ Mathematica. Time used: 0.044 (sec). Leaf size: 62

ode={D[x[t],{t,2}]+D[x[t],t]+D[y[t],t]-2*y[t]==40*Exp[3*t],D[x[t],t]+x[t]-D[y[t],t]==36*Exp[t]}; 
ic={x[0]==1,y[0]==3,Derivative[1][x][0] == 1}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to e^{-2 t} \left (e^{3 t} (9-6 t)-33 e^t+3 e^{5 t}+22\right ) \\ y(t)\to -12 e^t (t+1)+11 e^{-2 t}+4 e^{3 t} \\ \end{align*}

✓ Sympy. Time used: 0.302 (sec). Leaf size: 66

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-2*y(t) - 40*exp(3*t) + Derivative(x(t), t) + Derivative(x(t), (t, 2)) + Derivative(y(t), t),0),Eq(x(t) - 36*exp(t) + Derivative(x(t), t) - Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - C_{1} e^{- t} + 2 C_{2} e^{- 2 t} - 6 t e^{t} + \left (\frac {C_{3}}{2} + 11\right ) e^{t} + 3 e^{3 t}, \ y{\left (t \right )} = C_{2} e^{- 2 t} - 12 t e^{t} + \left (C_{3} - 8\right ) e^{t} + 4 e^{3 t}\right ] \]