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

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

when [tex]Q_0= 50[/tex]

"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 realise?

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.

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# Long-term behaviour of solution to ODE: oscillatory

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