76.9.10 problem 10

Internal problem ID [17441]
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 : 10
Date solved : Thursday, March 13, 2025 at 10:08:26 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=\frac {5 x \left (t \right )}{4}+\frac {3 y \left (t \right )}{4}\\ \frac {d}{d t}y \left (t \right )&=-\frac {3 x \left (t \right )}{4}-\frac {y \left (t \right )}{4} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 28

ode:=[diff(x(t),t) = 5/4*x(t)+3/4*y(t), diff(y(t),t) = -3/4*x(t)-1/4*y(t)]; 
ic:=x(0) = 2y(0) = 3; 
dsolve([ode,ic]);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{\frac {t}{2}} \left (\frac {15 t}{4}+2\right ) \\ y \left (t \right ) &= -\frac {{\mathrm e}^{\frac {t}{2}} \left (\frac {45 t}{4}-9\right )}{3} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 40

ode={D[x[t],t]==5/4*x[t]+3/4*y[t],D[y[t],t]==-3/4*x[t]-1/4*y[t]}; 
ic={x[0]==2,y[0]==3}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {1}{4} e^{t/2} (15 t+8) \\ y(t)\to -\frac {3}{4} e^{t/2} (5 t-4) \\ \end{align*}

✓ Sympy. Time used: 0.112 (sec). Leaf size: 51

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-5*x(t)/4 - 3*y(t)/4 + Derivative(x(t), t),0),Eq(3*x(t)/4 + y(t)/4 + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \frac {3 C_{2} t e^{\frac {t}{2}}}{4} + \left (\frac {3 C_{1}}{4} + C_{2}\right ) e^{\frac {t}{2}}, \ y{\left (t \right )} = - \frac {3 C_{1} e^{\frac {t}{2}}}{4} - \frac {3 C_{2} t e^{\frac {t}{2}}}{4}\right ] \]