72.18.3 problem 3

Internal problem ID [14885]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 4. Forcing and Resonance. Section 4.3 page 424
Problem number : 3
Date solved : Monday, March 31, 2025 at 01:01:22 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=-\cos \left (\frac {t}{2}\right ) \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 23

ode:=diff(diff(y(t),t),t)+4*y(t) = -cos(1/2*t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \sin \left (2 t \right ) c_2 +\cos \left (2 t \right ) c_1 -\frac {4 \cos \left (\frac {t}{2}\right )}{15} \]

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 30

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

\[ y(t)\to -\frac {4}{15} \cos \left (\frac {t}{2}\right )+c_1 \cos (2 t)+c_2 \sin (2 t) \]

✓ Sympy. Time used: 0.073 (sec). Leaf size: 24

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

\[ y{\left (t \right )} = C_{1} \sin {\left (2 t \right )} + C_{2} \cos {\left (2 t \right )} - \frac {4 \cos {\left (\frac {t}{2} \right )}}{15} \]