# Direct substitution in the analysis of periodic motion

1. Dec 9, 2008

### gsmith12

1. The problem statement, all variables and given/known data

By direct substitution, show that equation (3) is a solution of the differential equation (2).

2. Relevant equations

(2) (d^2 θ)/(dt^2 )=-g/l θ (Second derivative of θ(t)=-g/l θ.)

(3) θ(t)=θ_0 cos⁡(√(g/l) t)

3. The attempt at a solution

I tried to integrate equation (2) and derive equation (3) but it didn't come out correctly.

2. Dec 9, 2008

### tiny-tim

Welcome to PF!

Hi gsmith12! Welcome to PF!

i] The question doesn't want you to integrate …

it says use direct substitution … which means simply put θ(t) = θ0cos⁡(√(g/l) t) into (d^2 θ)/(dt^2 ), and show that it comes out as -g/l θ

ii] but if you still want to integrate, multiply both sides by dθ/dt first

3. Dec 9, 2008

### gsmith12

Thanks for the help. I think I am on the right track but am still running into a bit a difficulty .

To plug equation (3) into equation (2) do I first need to solve for theta? I am sorry but I am not completely clear on the set up for the direct substitution.

Thanks

4. Dec 10, 2008

### tiny-tim

welcome to reality!

Hi gsmith12!

I think you're slightly in denial about reality …
is a solution …

equation (3) has solved for θ.

So just plug-and-play!