52.10.4 problem 4

Internal problem ID [8398]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 8 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS. EXERCISES 8.2. Page 346
Problem number : 4
Date solved : Sunday, March 30, 2025 at 01:00:47 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-\frac {5 x \left (t \right )}{2}+2 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=\frac {3 x \left (t \right )}{4}-2 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.123 (sec). Leaf size: 35

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

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{-\frac {7 t}{2}}+c_2 \,{\mathrm e}^{-t} \\ y \left (t \right ) &= -\frac {c_1 \,{\mathrm e}^{-\frac {7 t}{2}}}{2}+\frac {3 c_2 \,{\mathrm e}^{-t}}{4} \\ \end{align*}

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 165

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

\begin{align*} x(t)\to \frac {1}{210} e^{\frac {1}{4} \left (t-\sqrt {105} t\right )} \left (3 c_1 \left (\left (35+3 \sqrt {105}\right ) e^{\frac {\sqrt {105} t}{2}}+35-3 \sqrt {105}\right )+8 \sqrt {105} c_2 \left (e^{\frac {\sqrt {105} t}{2}}-1\right )\right ) \\ y(t)\to \frac {1}{70} e^{\frac {1}{4} \left (t-\sqrt {105} t\right )} \left (\sqrt {105} c_1 \left (e^{\frac {\sqrt {105} t}{2}}-1\right )-c_2 \left (\left (3 \sqrt {105}-35\right ) e^{\frac {\sqrt {105} t}{2}}-35-3 \sqrt {105}\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.097 (sec). Leaf size: 36

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(5*x(t)/2 - 2*y(t) + Derivative(x(t), t),0),Eq(-3*x(t)/4 + 2*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - 2 C_{1} e^{- \frac {7 t}{2}} + \frac {4 C_{2} e^{- t}}{3}, \ y{\left (t \right )} = C_{1} e^{- \frac {7 t}{2}} + C_{2} e^{- t}\right ] \]