87.21.7 problem 7

Internal problem ID [23721]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 3. Linear Systems. Exercise at page 178
Problem number : 7
Date solved : Thursday, October 02, 2025 at 09:44:35 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }&=4 x-2 y \left (t \right )\\ y^{\prime }\left (t \right )&=x+y \left (t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.048 (sec). Leaf size: 31

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

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 32

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

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

✓ Sympy. Time used: 0.088 (sec). Leaf size: 26

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-4*x(t) + 2*y(t) + Derivative(x(t), t),0),Eq(-x(t) - y(t) + Derivative(y(t), t),0)] 
ics = {x(0): 1, y(0): 0} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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