70.9.3 problem 3

Internal problem ID [18800]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 3. Systems of two first order equations. Section 3.5 (Repeated Eigenvalues). Problems at page 188
Problem number : 3
Date solved : Thursday, October 02, 2025 at 03:31:00 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.120 (sec). Leaf size: 31

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

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 54

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

\begin{align*} x(t)&\to \frac {1}{2} e^{-t} (2 c_2 t-c_1 (t-2))\\ y(t)&\to \frac {1}{4} e^{-t} (c_1 (-t)+2 c_2 t+4 c_2) \end{align*}

✓ Sympy. Time used: 0.055 (sec). Leaf size: 39

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

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