78.5.18 problem 7.f

Internal problem ID [21057]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 6, Linear systems. Problems section 6.9
Problem number : 7.f
Date solved : Thursday, October 02, 2025 at 07:01:50 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.142 (sec). Leaf size: 58

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

\begin{align*} x \left (t \right ) &= {\mathrm e}^{2 t} \left (\sin \left (2 t \right ) c_1 +\cos \left (2 t \right ) c_2 \right ) \\ y \left (t \right ) &= -\frac {{\mathrm e}^{2 t} \left (\sin \left (2 t \right ) c_1 +2 \sin \left (2 t \right ) c_2 -2 \cos \left (2 t \right ) c_1 +\cos \left (2 t \right ) c_2 \right )}{5} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 67

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

\begin{align*} x(t)&\to \frac {1}{2} e^{2 t} (2 c_1 \cos (2 t)+(c_1+5 c_2) \sin (2 t))\\ y(t)&\to \frac {1}{2} e^{2 t} (2 c_2 \cos (2 t)-(c_1+c_2) \sin (2 t)) \end{align*}

✓ Sympy. Time used: 0.067 (sec). Leaf size: 58

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

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