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 : Tuesday, January 28, 2025 at 02:39:21 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime \prime }\left (t \right )+2 x \left (t \right )-2 y^{\prime }&=0\\ 3 x^{\prime }\left (t \right )+y^{\prime \prime }-8 y&=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\\ y^{\prime }\left (0\right ) = 0 \end{align*}

✓ Solution by Maple

Time used: 0.052 (sec). Leaf size: 65

dsolve([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), x(0) = 0, y(0) = 0, D(x)(0) = 0, D(y)(0) = 0], singsol=all)
 

\begin{align*} x &= -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 &= -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*}

✓ Solution by Mathematica

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

DSolve[{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]},{x[0]==0,y[0]==0,Derivative[1][x][0] == 0,Derivative[1][y][0] == 0},{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*}