28.4.26 problem 7.26

Internal problem ID [4558]
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 : 7.26
Date solved : Sunday, March 30, 2025 at 03:26:14 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d^{2}}{d t^{2}}x \left (t \right )+2 x \left (t \right )-2 \frac {d}{d t}y \left (t \right )&=0\\ 3 \frac {d}{d t}x \left (t \right )+\frac {d^{2}}{d t^{2}}y \left (t \right )-8 y \left (t \right )&=240 \,{\mathrm e}^{t} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.213 (sec). Leaf size: 65

ode:=[diff(diff(x(t),t),t)+2*x(t)-2*diff(y(t),t) = 0, 3*diff(x(t),t)+diff(diff(y(t),t),t)-8*y(t) = 240*exp(t)]; 
ic:=x(0) = 0y(0) = 0D(x)(0) = 0D(y)(0) = 0; 
dsolve([ode,ic]);
 

\begin{align*} x \left (t \right ) &= -32 \,{\mathrm e}^{t}+12 \cos \left (2 t \right )-10 \,{\mathrm e}^{-2 t}+30 \,{\mathrm e}^{2 t}-24 \sin \left (2 t \right ) \\ y \left (t \right ) &= -48 \,{\mathrm e}^{t}-6 \sin \left (2 t \right )+15 \,{\mathrm e}^{-2 t}+45 \,{\mathrm e}^{2 t}-12 \cos \left (2 t \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.414 (sec). Leaf size: 72

ode={D[x[t],{t,2}]+2*x[t]-2*D[y[t],t]==0,3*D[x[t],t]+D[y[t],{t,2}]-8*y[t]==240*Exp[t]}; 
ic={x[0]==0,y[0]==0,Derivative[1][x][0] == 0,Derivative[1][y][0] == 0}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to -10 e^{-2 t}-32 e^t+30 e^{2 t}-24 \sin (2 t)+12 \cos (2 t) \\ y(t)\to 15 e^{-2 t}-48 e^t+45 e^{2 t}-6 \sin (2 t)-12 \cos (2 t) \\ \end{align*}

✓ Sympy. Time used: 0.376 (sec). Leaf size: 126

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(2*x(t) + Derivative(x(t), (t, 2)) - 2*Derivative(y(t), t),0),Eq(-8*y(t) - 240*exp(t) + 3*Derivative(x(t), t) + Derivative(y(t), (t, 2)),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \frac {C_{1} e^{- 2 t}}{3} + \frac {C_{2} e^{2 t}}{3} + C_{3} \sin {\left (2 t \right )} - C_{4} \cos {\left (2 t \right )} - 12 e^{t} \sin ^{2}{\left (2 t \right )} - 12 e^{t} \cos ^{2}{\left (2 t \right )} - 20 e^{t}, \ y{\left (t \right )} = - \frac {C_{1} e^{- 2 t}}{2} + \frac {C_{2} e^{2 t}}{2} + \frac {C_{3} \cos {\left (2 t \right )}}{2} + \frac {C_{4} \sin {\left (2 t \right )}}{2} + 12 e^{t} \sin ^{2}{\left (2 t \right )} + 12 e^{t} \cos ^{2}{\left (2 t \right )} - 60 e^{t}\right ] \]