82.55.5 problem Ex. 5

Internal problem ID [18969]
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 : Monday, March 31, 2025 at 06:26:54 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*}

✓ Maple. Time used: 0.107 (sec). Leaf size: 69

ode:=[diff(diff(x(t),t),t)-3*x(t)-4*y(t) = 0, diff(diff(y(t),t),t)+x(t)+y(t) = 0]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{t}+c_2 \,{\mathrm e}^{t} t +c_3 \,{\mathrm e}^{-t}+c_4 \,{\mathrm e}^{-t} t \\ y \left (t \right ) &= -\frac {c_1 \,{\mathrm e}^{t}}{2}-\frac {c_2 \,{\mathrm e}^{t} t}{2}+\frac {c_2 \,{\mathrm e}^{t}}{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*}

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

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

✓ Sympy. Time used: 0.176 (sec). Leaf size: 61

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-3*x(t) - 4*y(t) + Derivative(x(t), (t, 2)),0),Eq(x(t) + y(t) + Derivative(y(t), (t, 2)),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = 2 C_{1} t e^{t} - 2 C_{2} e^{- t} - 2 C_{3} t e^{- t} + 2 C_{4} e^{t}, \ y{\left (t \right )} = - C_{1} t e^{t} + C_{3} t e^{- t} + \left (C_{1} - C_{4}\right ) e^{t} + \left (C_{2} + C_{3}\right ) e^{- t}\right ] \]