76.9.9 problem 9

Internal problem ID [17440]
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 : 9
Date solved : Thursday, March 13, 2025 at 10:08:24 AM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x \left (0\right ) = 3\\ y \left (0\right ) = -2 \end{align*}

✓ Maple. Time used: 0.069 (sec). Leaf size: 28

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

\begin{align*} x \left (t \right ) &= {\mathrm e}^{\frac {t}{2}} \left (\frac {3 t}{2}+3\right ) \\ y \left (t \right ) &= -\frac {{\mathrm e}^{\frac {t}{2}} \left (\frac {9 t}{2}+6\right )}{3} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 38

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

\begin{align*} x(t)\to \frac {3}{2} e^{t/2} (t+2) \\ y(t)\to -\frac {1}{2} e^{t/2} (3 t+4) \\ \end{align*}

✓ Sympy. Time used: 0.115 (sec). Leaf size: 51

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

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