70.9.2 problem 2

Internal problem ID [18799]
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 : 2
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 {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*}

✓ Maple. Time used: 0.113 (sec). Leaf size: 34

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)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{\frac {t}{2}} \left (c_2 t +c_1 \right ) \\ y \left (t \right ) &= -\frac {{\mathrm e}^{\frac {t}{2}} \left (3 c_2 t +3 c_1 -4 c_2 \right )}{3} \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 57

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={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{4} e^{t/2} (c_1 (3 t+4)+3 c_2 t)\\ y(t)&\to \frac {1}{4} e^{t/2} (4 c_2-3 (c_1+c_2) t) \end{align*}

✓ Sympy. Time used: 0.066 (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)
 

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