82.55.5 problem Ex. 5

Internal problem ID [19040]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter XI. Ordinary differential equations with more than two variables. problems at page 129
Problem number : Ex. 5
Date solved : Tuesday, January 28, 2025 at 08:29:27 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d^{2}}{d t^{2}}x \left (t \right )-3 x \left (t \right )-4 y \left (t \right )&=0\\ \frac {d^{2}}{d t^{2}}y \left (t \right )+x \left (t \right )+y \left (t \right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.026 (sec). Leaf size: 69

dsolve([diff(x(t),t$2)-3*x(t)-4*y(t)=0,diff(y(t),t$2)+x(t)+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 {{\mathrm e}^{-t} c_{2}}{2}-\frac {c_3 \,{\mathrm e}^{t}}{2}-\frac {c_4 \,{\mathrm e}^{t} t}{2}+\frac {c_4 \,{\mathrm e}^{t}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.014 (sec). Leaf size: 169

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