43.6.11 problem 4(c)

Internal problem ID [8925]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 2. Linear equations with constant coefficients. Page 69
Problem number : 4(c)
Date solved : Tuesday, September 30, 2025 at 06:00:13 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+\omega ^{2} y&=A \cos \left (\omega x \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 18

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

\[ y = \frac {\sin \left (\omega x \right ) \left (1+\frac {A x}{2}\right )}{\omega } \]

✓ Mathematica. Time used: 0.035 (sec). Leaf size: 75

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

\begin{align*} y(x)&\to -\frac {\sin (x \omega ) \left (-4 \omega ^2 \int _1^x\frac {A \cos ^2(\omega K[1])}{\omega }dK[1]+4 \omega ^2 \int _1^0\frac {A \cos ^2(\omega K[1])}{\omega }dK[1]+A \sin (2 x \omega )-4 \omega \right )}{4 \omega ^2} \end{align*}

✓ Sympy. Time used: 0.092 (sec). Leaf size: 37

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

\[ y{\left (x \right )} = \frac {A x \sin {\left (\omega x \right )}}{2 \omega } - \frac {i e^{i \omega x}}{2 \omega } + \frac {i e^{- i \omega x}}{2 \omega } \]