Is Substituting q(t) the Correct Method to Verify a Differential Equation?

[WWCY]

Homework Statement

How does one show that q(t) is indeed a solution?

Homework Equations

The Attempt at a Solution

My current idea is that i should come up with any form of solution, like q = Acos(ωt), and slot it in the RHS.
Reason being that if q is indeed a solution, the result of the substitution should look something like the RHS.

Am I missing something? Thanks in advance.

 

[S.G. Janssens]

Am I missing something?

I would read the question differently than you did.

I think that this is a modeling question, because no expression for $q$ is given yet. What I would do, is show from basic circuit principles that the dynamics of the charge as a function of time is modeled by the given differential equation.

Only in the next step(s), you will then probably be asked to (or led to) find the solution of that DE.

 

[WWCY]

What I would do, is show from basic circuit principles that the dynamics of the charge as a function of time is modeled by the given differential equation.

Thanks for replying!

What did you mean by the above statement? Could you elaborate a little?

Also, I found this bit of proof in relation to these types of DEs in a calculus text:

Could this be another way to approach the problem?

 

[Ray Vickson]

Homework Statement

View attachment 209296
How does one show that q(t) is indeed a solution?

Homework Equations

The Attempt at a Solution

My current idea is that i should come up with any form of solution, like q = Acos(ωt), and slot it in the RHS.
Reason being that if q is indeed a solution, the result of the substitution should look something like the RHS.

Am I missing something? Thanks in advance.

No, as Krylov has said, you have mis-read the question. The question said "Show that $q(t)$ obeys ... " so it is not asking for a solution of a differential equation. It is asking you to prove that the given differential equation is a correct formulation.

The point is: first you get the correct DE; then you worry about solving it.

 

[WWCY]

No, as Krylov has said, you have mis-read the question. The question said "Show that $q(t)$ obeys ... " so it is not asking for a solution of a differential equation. It is asking you to prove that the given differential equation is a correct formulation.

The point is: first you get the correct DE; then you worry about solving it.

Hi, thanks for replying.

So what you mean is that I should first derive the equation? (Apologies if I was slow to understand this)

But say I've already derived the equation, do I still need to check whether or not it's "correct" by substituting q = q_p + q_c?

Thank you.