75.27.10 problem 785

Internal problem ID [17175]
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 : 785
Date solved : Monday, March 31, 2025 at 03:43:54 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.113 (sec). Leaf size: 46

ode:=[diff(diff(x(t),t),t) = 3*x(t)+y(t), diff(y(t),t) = -2*x(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 125

ode={D[x[t],{t,2}]==3*x[t]+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}{9} e^{-2 t} \left (c_1 \left (e^{3 t} (3 t+5)+4\right )+c_2 \left (e^{3 t} (3 t+2)-2\right )+c_3 \left (e^{3 t} (3 t-1)+1\right )\right ) \\ y(t)\to \frac {1}{9} e^{-2 t} \left (c_1 \left (4-2 e^{3 t} (3 t+2)\right )+c_2 \left (e^{3 t} (2-6 t)-2\right )+c_3 \left (e^{3 t} (8-6 t)+1\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.140 (sec). Leaf size: 48

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

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