75.27.3 problem 778

Internal problem ID [17160]
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 : 778
Date solved : Thursday, March 13, 2025 at 09:18:04 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )+3 x \left (t \right )+4 y \left (t \right )&=0\\ \frac {d}{d t}y \left (t \right )+2 x \left (t \right )+5 y \left (t \right )&=0 \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 1\\ y \left (0\right ) = 4 \end{align*}

✓ Maple. Time used: 0.038 (sec). Leaf size: 31

ode:=[diff(x(t),t)+3*x(t)+4*y(t) = 0, diff(y(t),t)+2*x(t)+5*y(t) = 0]; 
ic:=x(0) = 1y(0) = 4; 
dsolve([ode,ic]);
 

\begin{align*} x \left (t \right ) &= 3 \,{\mathrm e}^{-7 t}-2 \,{\mathrm e}^{-t} \\ y \left (t \right ) &= 3 \,{\mathrm e}^{-7 t}+{\mathrm e}^{-t} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 36

ode={D[x[t],t]+3*x[t]+4*y[t]==0,D[y[t],t]+2*x[t]+5*y[t]==0}; 
ic={x[0]==1,y[0]==4}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to e^{-7 t} \left (3-2 e^{6 t}\right ) \\ y(t)\to e^{-7 t} \left (e^{6 t}+3\right ) \\ \end{align*}

✓ Sympy. Time used: 0.098 (sec). Leaf size: 29

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(3*x(t) + 4*y(t) + Derivative(x(t), t),0),Eq(2*x(t) + 5*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = C_{1} e^{- 7 t} - 2 C_{2} e^{- t}, \ y{\left (t \right )} = C_{1} e^{- 7 t} + C_{2} e^{- t}\right ] \]