Construct DE w/ Soln y(t) = e^t cos(3t)

[LocalStudent]

Hi

I'm working through some example question and the memo leaves out this question. I was hoping someone can help me out.

The question:
Construct a differential equation of the form y'' + by' + cy = 0 which has y(t) = e^t cos(3t) as one of its solutions.

What I did:
First I found the y'(t) and y''(t)
Plugged that into the DE.

My answer:
b = -2
c = -6

y'' - 2y' - 6y = 0

 

[Simon Bridge]

Check your answer by plugging the solution into it from scratch.
It will also help to show your working step-by-step.

 

[mfb]

You can use WolframAlpha to check your result.

 

[pasmith]

Hi

I'm working through some example question and the memo leaves out this question. I was hoping someone can help me out.

The question:
Construct a differential equation of the form y'' + by' + cy = 0 which has y(t) = e^t cos(3t) as one of its solutions.

What I did:
First I found the y'(t) and y''(t)
Plugged that into the DE.

My answer:
b = -2
c = -6

y'' - 2y' - 6y = 0

Hint: What is the relationship between the ODE
$$y'' + by' + cy = 0$$
and the quadratic equation
$$\lambda^2 + b\lambda + c = 0?$$

You may also find the identity
$$\cos kt \equiv \frac{e^{ikt} + e^{-ikt}}{2}$$
helpful.