76.19.7 problem 7

Internal problem ID [17652]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 5. The Laplace transform. Section 5.4 (Solving differential equations with Laplace transform). Problems at page 327
Problem number : 7
Date solved : Thursday, March 13, 2025 at 10:45:59 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Maple. Time used: 10.907 (sec). Leaf size: 27

ode:=diff(diff(y(t),t),t)+w^2*y(t) = cos(2*t); 
ic:=y(0) = 1, D(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \frac {\cos \left (2 t \right )+\cos \left (w t \right ) \left (w^{2}-5\right )}{w^{2}-4} \]

✓ Mathematica. Time used: 0.18 (sec). Leaf size: 28

ode=D[y[t],{t,2}]+w^2*y[t]==Cos[2*t]; 
ic={y[0]==1,Derivative[1][y][0] == 0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {\left (w^2-5\right ) \cos (t w)+\cos (2 t)}{w^2-4} \]

✓ Sympy. Time used: 0.134 (sec). Leaf size: 49

from sympy import * 
t = symbols("t") 
w = symbols("w") 
y = Function("y") 
ode = Eq(w**2*y(t) - cos(2*t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {\left (w^{2} - 5\right ) e^{i t w}}{2 w^{2} - 8} + \frac {\left (w^{2} - 5\right ) e^{- i t w}}{2 w^{2} - 8} + \frac {\cos {\left (2 t \right )}}{w^{2} - 4} \]