14.7.13 problem Problem 38

Internal problem ID [3728]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 8, Linear differential equations of order n. Section 8.3, The Method of Undetermined Coefficients. page 525
Problem number : Problem 38
Date solved : Tuesday, September 30, 2025 at 06:56:25 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+\omega ^{2} y&=\frac {F_{0} \cos \left (\omega t \right )}{m} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 23

ode:=diff(diff(y(t),t),t)+omega^2*y(t) = F__0/m*cos(omega*t); 
ic:=[y(0) = 1, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = \cos \left (\omega t \right )+\frac {F_{0} \sin \left (\omega t \right ) t}{2 \omega m} \]

✓ Mathematica. Time used: 0.04 (sec). Leaf size: 26

ode=D[y[t],{t,2}]+\[Omega]^2*y[t]==F0/m*Cos[\[Omega]*t]; 
ic={y[0]==1,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {\text {F0} t \sin (t \omega )}{2 m \omega }+\cos (t \omega ) \end{align*}

✓ Sympy. Time used: 0.103 (sec). Leaf size: 34

from sympy import * 
t = symbols("t") 
F__0 = symbols("F__0") 
m = symbols("m") 
omega = symbols("omega") 
y = Function("y") 
ode = Eq(-F__0*cos(omega*t)/m + omega**2*y(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)
 

\[ y{\left (t \right )} = \frac {F^{0} t \sin {\left (\omega t \right )}}{2 m \omega } + \frac {e^{i \omega t}}{2} + \frac {e^{- i \omega t}}{2} \]