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

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

✓ Solution by Maple

Time used: 0.059 (sec). Leaf size: 57

dsolve([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), x(0) = 1, y(0) = 3, D(x)(0) = 1], singsol=all)
 

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

✓ Solution by Mathematica

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

DSolve[{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]},{x[0]==1,y[0]==3,Derivative[1][x][0] == 1},{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*}