28.4.5 problem 7.5

Internal problem ID [4537]
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 : 7.5
Date solved : Tuesday, January 28, 2025 at 02:39:20 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime \prime }\left (t \right )-3 x \left (t \right )-4 y&=0\\ x \left (t \right )+y^{\prime \prime }+y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.038 (sec). Leaf size: 69

dsolve([diff(x(t),t$2)-3*x(t)-4*y(t)=0,x(t)+diff(y(t),t$2)+y(t)=0],singsol=all)
 

\begin{align*} x &= {\mathrm e}^{t} c_{1} +c_{2} {\mathrm e}^{t} t +c_3 \,{\mathrm e}^{-t}+c_4 \,{\mathrm e}^{-t} t \\ y &= -\frac {{\mathrm e}^{t} c_{1}}{2}-\frac {c_{2} {\mathrm e}^{t} t}{2}-\frac {c_3 \,{\mathrm e}^{-t}}{2}-\frac {c_4 \,{\mathrm e}^{-t} t}{2}+\frac {c_{2} {\mathrm e}^{t}}{2}-\frac {{\mathrm e}^{-t} c_4}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.016 (sec). Leaf size: 169

DSolve[{D[x[t],{t,2}]-3*x[t]-4*y[t]==0,x[t]+D[y[t],{t,2}]+y[t]==0},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{2} e^{-t} \left (c_1 \left (-t+e^{2 t} (t+1)+1\right )-2 c_4 \left (e^{2 t}-1\right )+t \left (c_2 \left (e^{2 t}+1\right )+2 c_3 \left (e^{2 t}-1\right )+2 c_4 \left (e^{2 t}+1\right )\right )\right ) \\ y(t)\to \frac {1}{4} e^{-t} \left (c_2 \left (e^{2 t}-1\right )+2 c_3 \left (e^{2 t}+1\right )+4 c_4 \left (e^{2 t}-1\right )-t \left (c_1 \left (e^{2 t}-1\right )+c_2 \left (e^{2 t}+1\right )+2 c_3 \left (e^{2 t}-1\right )+2 c_4 \left (e^{2 t}+1\right )\right )\right ) \\ \end{align*}