Long-term behaviour of solution to ODE: oscillatory

[jellicorse]

Homework Statement

I wondered if anyone could advise me how to proceed with this question.

The solution to the differential equation $$\frac{dQ}{dt}= \frac{1}{2}+\frac{1}{4}sin(t)-\frac{Q}{50}$$ is $$Q=25+(\frac{sin(t)-625cos(t)+63150e^{-\frac{t}{50}}}{2501})$$

when $$Q_0= 50$$

"The long-term behaviour of the solution is an oscillation about a certain constant level. What is this level? What is the amplitude of oscillation?"

I have plotted the solution and can see that it tails off and oscillates around what looks like a level of Q=25.

Would I be right in saying the oscillation's minimum will be Q=25, as this is the least value the expression for Q can realize?


My biggest problem is finding the amplitude of oscillation. I can see that the term involving 'e' at the end of the expression for Q goes to zero for large 't'. But I am not sure how to obtain the amplitude from what is left.

 

[LCKurtz]

The amplitude of $A\cos(\omega t) + B\sin(\omega t)$ is $\sqrt{A^2+B^2}$.

 

[pasmith]

Homework Statement

I wondered if anyone could advise me how to proceed with this question.

The solution to the differential equation $$\frac{dQ}{dt}= \frac{1}{2}+\frac{1}{4}sin(t)-\frac{Q}{50}$$ is $$Q=25+(\frac{sin(t)-625cos(t)+63150e^{-\frac{t}{50}}}{2501})$$

when $$Q_0= 50$$

"The long-term behaviour of the solution is an oscillation about a certain constant level. What is this level? What is the amplitude of oscillation?"

I have plotted the solution and can see that it tails off and oscillates around what looks like a level of Q=25.

Would I be right in saying the oscillation's minimum will be Q=25, as this is the least value the expression for Q can realize?

No, 25 is the value about which the solution oscillates in the long term.

My biggest problem is finding the amplitude of oscillation. I can see that the term involving 'e' at the end of the expression for Q goes to zero for large 't'. But I am not sure how to obtain the amplitude from what is left.

Compare the oscillatory part of your solution with the right hand side of the angle sum formula
$$R\cos(\phi + t) = R\cos\phi\cos t - R\sin\phi \sin t$$
and hence determine $R > 0$ (and $\phi$, which you don't actually need here).

 

[jellicorse]

Thanks LCKurtz and pasmith... I need to revisit those angle sum formulas for a bit.

 

[epenguin]

1. I have plotted the solution and can see that it tails off and oscillates around what looks like a level of Q=25.

2. Would I be right in saying the oscillation's minimum will be Q=25, as this is the least value the expression for Q can realize?


3. My biggest problem is finding the amplitude of oscillation. I can see that the term involving 'e' at the end of the expression for Q goes to zero for large 't'. But I am not sure how to obtain the amplitude from what is left.

1 you should be to see that at a glance at the formulae without plotting.

2. It oscillates about 25 as was already pointed out - have you copied the question correctly as it would be surprising if this were not asked? Especially as asking the minimum and asking the amplitude is practically the same question.

3. You should immediately realize that tha amplitude is going to be very little different from 625/2501 ≈ 1/4. In fact it seems to me that 626/2501 should be a very good approximation, the angle where the maximum occurs being approximately 1/626 radians.

 

[jellicorse]

Yes, the question did ask around what value it oscillates... After a bit of revision on the basic trig, I think I can see how this works now. Thanks.