28.4.10 problem 7.10

Internal problem ID [4542]
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 : 7.10
Date solved : Monday, January 27, 2025 at 09:23:16 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )-x \left (t \right )-2 y&={\mathrm e}^{t}\\ -4 x \left (t \right )+y^{\prime }-3 y&=1 \end{align*}

✓ Solution by Maple

Time used: 0.058 (sec). Leaf size: 45

dsolve([diff(x(t),t)-x(t)-2*y(t)=exp(t),-4*x(t)+diff(y(t),t)-3*y(t)=1],singsol=all)
 

\begin{align*} x &= \frac {c_{2} {\mathrm e}^{5 t}}{2}-{\mathrm e}^{-t} c_{1} +\frac {{\mathrm e}^{t}}{4}-\frac {2}{5} \\ y &= c_{2} {\mathrm e}^{5 t}+{\mathrm e}^{-t} c_{1} -\frac {{\mathrm e}^{t}}{2}+\frac {1}{5} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.077 (sec). Leaf size: 91

DSolve[{D[x[t],t]-x[t]-2*y[t]==Exp[t],-4*x[t]+D[y[t],t]-3*y[t]==1},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{60} e^{-t} \left (-24 e^t+15 e^{2 t}+20 (c_1+c_2) e^{6 t}+40 c_1-20 c_2\right ) \\ y(t)\to -\frac {e^t}{2}+\frac {1}{3} (c_2-2 c_1) e^{-t}+\frac {2}{3} (c_1+c_2) e^{5 t}+\frac {1}{5} \\ \end{align*}