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 : Monday, January 27, 2025 at 09:23:20 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.088 (sec). Leaf size: 71

dsolve([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],singsol=all)
 

\begin{align*} x &= c_{2} \sin \left (2 t \right )+c_{1} \cos \left (2 t \right )+2 \,{\mathrm e}^{t}-3 t +5+\frac {3 \sin \left (2 t \right )}{2}+5 \cos \left (2 t \right ) \\ y &= c_{2} \cos \left (2 t \right )-c_{1} \sin \left (2 t \right )+{\mathrm e}^{t}+\frac {3 \cos \left (2 t \right )}{2}-5 \sin \left (2 t \right )-t \\ \end{align*}

✓ Solution by Mathematica

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

DSolve[{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},{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*}