Finding the resonance of a second order differential equation

Question

$$y'' + 100y = \frac{1}{3} \cos(bx) $$

with $$ y(0) = 0$$ and $$ y'(0) = 0.1$$

I believe the homogenous (general) solution is $$\frac{1}{100} \sin(10x)$$

however I am having trouble finding the inhomogeneous (particular) solution as well as the values of b at which resonance will occur (I think these two are intertwined).

score 1 · Answer 1 · answered Oct 15 '19 at 09:21

1

We have that for

$$y''+\omega_0^2y=0 \implies y=A\cos(\omega_0 t+\phi)$$

therefore in that case $\omega_0=10$ and resonance occurs for $b=10$.

answered Oct 15 '19 at 09:21

user

154,566

score 1 · Answer 2 · answered Oct 15 '19 at 09:22

1

The general solution to the homogeneous equation is $A\sin (10x)+B\cos (10x)$. A particular solution to the inhomogeneous equation is $c \cos (bx)$ for a suitable $c$. Can you take it from here?

answered Oct 15 '19 at 09:22

Kavi Rama Murthy

311,013

Why is it c cos(bx) and not c*x cos (bx) because of linear dependence? – J. Cricks Oct 15 '19 at 10:42
I found a particular solution by just guess work. $\cos (bx)$ is not a solution by itself but constant times this function is a solution. – Kavi Rama Murthy Oct 15 '19 at 11:47
@J.Cricks For finding particular solution it is always advisable to see if guesswork yields one. You don't always need a technique for finding a particular solution – Kavi Rama Murthy Oct 15 '19 at 12:03
can there me multiple values of b for which resonance can occur? – J. Cricks Oct 18 '19 at 07:34

score 0 · Answer 3 · answered Oct 15 '19 at 09:30

0

The resonance is a plant characteristic. The plant has transfer function $G(s) = \frac{1}{s^2+10^2}$ so the resonance occurs when the stimulating frequency is $b = 10$

answered Oct 15 '19 at 09:30

Cesareo

33,252

Finding the resonance of a second order differential equation

3 Answers3