75.29.5 problem 806

Internal problem ID [17180]
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 22. Integration of homogeneous linear systems with constant coefficients. Eulers method. Exercises page 230
Problem number : 806
Date solved : Thursday, March 13, 2025 at 09:18:20 AM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Maple. Time used: 0.046 (sec). Leaf size: 27

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

\begin{align*} x \left (t \right ) &= -5 \,{\mathrm e}^{2 t} \sin \left (t \right ) \\ y \left (t \right ) &= {\mathrm e}^{2 t} \left (\cos \left (t \right )-2 \sin \left (t \right )\right ) \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 27

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

\begin{align*} x(t)\to -4 e^{2 t} t \\ y(t)\to e^{2 t} (1-2 t) \\ \end{align*}

✓ Sympy. Time used: 0.120 (sec). Leaf size: 53

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

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