5.3.8 problem 15

Internal problem ID [1490]
Book : Elementary differential equations and boundary value problems, 11th ed., Boyce, DiPrima, Meade
Section : Chapter 6.2, The Laplace Transform. Solution of Initial Value Problems. page 255
Problem number : 15
Date solved : Tuesday, September 30, 2025 at 04:34:36 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+\omega ^{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: 0.173 (sec). Leaf size: 27

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

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

✓ Mathematica. Time used: 0.116 (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]
 

\begin{align*} y(t)&\to \frac {\left (w^2-5\right ) \cos (t w)+\cos (2 t)}{w^2-4} \end{align*}

✓ Sympy. Time used: 0.089 (sec). Leaf size: 49

from sympy import * 
t = symbols("t") 
omega = symbols("omega") 
y = Function("y") 
ode = Eq(omega**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)
 

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