86.12.3 problem 3

Internal problem ID [23210]
Book : An introduction to Differential Equations. By Howard Frederick Cleaves. 1969. Oliver and Boyd publisher. ISBN 0050015044
Section : Chapter 10. The Laplace transform. Exercise 10c at page 156
Problem number : 3
Date solved : Thursday, October 02, 2025 at 09:24:24 PM
CAS classification : system_of_ODEs

\begin{align*} 4 \frac {d}{d t}x \left (t \right )-2 y \left (t \right )&=\cos \left (2 t \right )\\ x \left (t \right )-2 \frac {d}{d t}y \left (t \right )&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.100 (sec). Leaf size: 45

ode:=[4*diff(x(t),t)-2*y(t) = cos(2*t), x(t)-2*diff(y(t),t) = 0]; 
ic:=[x(0) = 0, y(0) = 0]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= \frac {{\mathrm e}^{\frac {t}{2}}}{68}-\frac {{\mathrm e}^{-\frac {t}{2}}}{68}+\frac {2 \sin \left (2 t \right )}{17} \\ y \left (t \right ) &= \frac {{\mathrm e}^{\frac {t}{2}}}{68}+\frac {{\mathrm e}^{-\frac {t}{2}}}{68}-\frac {\cos \left (2 t \right )}{34} \\ \end{align*}

✓ Mathematica. Time used: 0.121 (sec). Leaf size: 61

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

\begin{align*} x(t)&\to \frac {1}{68} \left (e^{-t/2} \left (e^t-1\right )+8 \sin (2 t)\right )\\ y(t)&\to \frac {1}{68} e^{-t/2} \left (e^t-2 e^{t/2} \cos (2 t)+1\right ) \end{align*}

✓ Sympy. Time used: 0.239 (sec). Leaf size: 49

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

\[ \left [ x{\left (t \right )} = \frac {e^{\frac {t}{2}}}{68} + \frac {2 \sin {\left (2 t \right )}}{17} - \frac {e^{- \frac {t}{2}}}{68}, \ y{\left (t \right )} = \frac {e^{\frac {t}{2}}}{68} - \frac {\cos {\left (2 t \right )}}{34} + \frac {e^{- \frac {t}{2}}}{68}\right ] \]