70.9.5 problem 5

Internal problem ID [18802]
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 : 5
Date solved : Thursday, October 02, 2025 at 03:31:01 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.111 (sec). Leaf size: 31

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

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 49

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

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

✓ Sympy. Time used: 0.054 (sec). Leaf size: 39

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

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