70.9.12 problem 12

Internal problem ID [18809]
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 : 12
Date solved : Thursday, October 02, 2025 at 03:31:05 PM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x \left (0\right )&=1 \\ y \left (0\right )&=3 \\ \end{align*}

✓ Maple. Time used: 0.128 (sec). Leaf size: 28

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

\begin{align*} x \left (t \right ) &= {\mathrm e}^{\frac {3 t}{2}} \left (2 t +1\right ) \\ y \left (t \right ) &= -{\mathrm e}^{\frac {3 t}{2}} \left (2 t -3\right ) \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 34

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

\begin{align*} x(t)&\to e^{3 t/2} (2 t+1)\\ y(t)&\to e^{3 t/2} (3-2 t) \end{align*}

✓ Sympy. Time used: 0.064 (sec). Leaf size: 51

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

\[ \left [ x{\left (t \right )} = \frac {C_{2} t e^{\frac {3 t}{2}}}{2} + \left (\frac {C_{1}}{2} + C_{2}\right ) e^{\frac {3 t}{2}}, \ y{\left (t \right )} = - \frac {C_{1} e^{\frac {3 t}{2}}}{2} - \frac {C_{2} t e^{\frac {3 t}{2}}}{2}\right ] \]