73.27.1 problem 38.1

Internal problem ID [15686]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 38. Systems of differential equations. A starting point. Additional Exercises. page 786
Problem number : 38.1
Date solved : Monday, March 31, 2025 at 01:45:05 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=2 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=1-2 x \left (t \right ) \end{align*}

✓ Maple. Time used: 0.130 (sec). Leaf size: 35

ode:=[diff(x(t),t) = 2*y(t), diff(y(t),t) = 1-2*x(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= \sin \left (2 t \right ) c_2 +\cos \left (2 t \right ) c_1 +\frac {1}{2} \\ y \left (t \right ) &= \cos \left (2 t \right ) c_2 -\sin \left (2 t \right ) c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 112

ode={D[x[t],t]==2*y[t],D[y[t],t]==1-2*x[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \cos (2 t) \int _1^t-\sin (2 K[1])dK[1]+\sin (2 t) \int _1^t\cos (2 K[2])dK[2]+c_1 \cos (2 t)+c_2 \sin (2 t) \\ y(t)\to -\sin (2 t) \int _1^t-\sin (2 K[1])dK[1]+\cos (2 t) \int _1^t\cos (2 K[2])dK[2]+c_2 \cos (2 t)-c_1 \sin (2 t) \\ \end{align*}

✓ Sympy. Time used: 0.118 (sec). Leaf size: 48

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-2*y(t) + Derivative(x(t), t),0),Eq(2*x(t) + Derivative(y(t), t) - 1,0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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