76.16.8 problem 23

Internal problem ID [17614]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.6 (Forced vibrations, Frequency response, and Resonance). Problems at page 272
Problem number : 23
Date solved : Monday, March 31, 2025 at 04:22:36 PM
CAS classification : [NONE]

\begin{align*} y^{\prime \prime }+y+\frac {y^{3}}{5}&=\cos \left (w t \right ) \end{align*}

With initial conditions

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

✗ Maple

ode:=diff(diff(y(t),t),t)+y(t)+1/5*y(t)^3 = cos(w*t); 
ic:=y(0) = 0, D(y)(0) = 0; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=D[y[t],{t,2}]+y[t]+2/10*y[t]^3==Cos[w*t]; 
ic={y[0]==0,Derivative[1][y][0] == 0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
t = symbols("t") 
w = symbols("w") 
y = Function("y") 
ode = Eq(y(t)**3/5 + y(t) - cos(t*w) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

NotImplementedError : solve: Cannot solve y(t)**3/5 + y(t) - cos(t*w) + Derivative(y(t), (t, 2))