75.27.8 problem 783

Internal problem ID [17165]
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 : 783
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 )&=y \left (t \right )\\ \frac {d^{2}}{d t^{2}}y \left (t \right )&=x \left (t \right ) \end{align*}

✓ Maple. Time used: 0.053 (sec). Leaf size: 47

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

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

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 172

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

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

✓ Sympy. Time used: 0.130 (sec). Leaf size: 44

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

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