75.27.9 problem 784

Internal problem ID [17166]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 3 (Systems of differential equations). Section 20. The method of elimination. Exercises page 212
Problem number : 784
Date solved : Friday, March 14, 2025 at 04:49:25 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.104 (sec). Leaf size: 39

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

\begin{align*} x \left (t \right ) &= c_{1} -\frac {1}{2} c_{1} t^{2}-c_{2} t -c_{3} \\ y \left (t \right ) &= \frac {1}{6} c_{1} t^{3}+\frac {1}{2} c_{2} t^{2}+c_{3} t +c_4 \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 61

ode={D[x[t],{t,2}]+D[y[t],t]+x[t]==0,D[x[t],t]+D[y[t],{t,2}]==0}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to -\frac {c_1 t^2}{2}-\frac {c_4 t^2}{2}+c_2 t+c_1 \\ y(t)\to \frac {1}{6} (c_1+c_4) t^3-\frac {c_2 t^2}{2}+c_4 t+c_3 \\ \end{align*}

✓ Sympy. Time used: 0.124 (sec). Leaf size: 36

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

\[ \left [ x{\left (t \right )} = - C_{1} - \frac {C_{2} t^{2}}{2} + C_{2} - C_{3} t, \ y{\left (t \right )} = C_{1} t + \frac {C_{2} t^{3}}{6} + \frac {C_{3} t^{2}}{2} + C_{4}\right ] \]