Differential Equations Question (1 Viewer)

notme123

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This question is giving me a headache starting from part a, I really dont even understand what theyre asking for...
Would anyone be able to please help?

Thankyou!
What is Py supposed to mean? Is it multiplication or a subscript? If it's a subscript then what's that notation?
 

Qeru

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Could you show what Q12 is so we actually know what the variables and constants mean?
 

CM_Tutor

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We have a function where , a constant, which satisfies the differential equation (DE)



where and are constants.

(a) Let be a function related to by the equation . It follows that



which can be substituted into (*) to give:



Now, let , a constant, and the DE becomes:



with the initial value of being

.

(b) Let be a variable related to time by , from which it follows that

.

From the Chain Rule, we know that



and we can substitute (**) and (1) into this to get



which is a DE in two variables, and . At , it has the initial value

.

(c) Let be a variable related to by



from which it follows that



The problem has now been transformed into one that can be solved by integration without the need to use partial fractions:




I'll leave the rest for you to work on. :)
 
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CM_Tutor

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Is it or , why can you factor out a P like that?
It is . The point is to take a DE that calls for partial fractions and get a solution without using that approach.

If it was then you are correct that would not be a common factor that could be separated. Also, we would need a definition of .
 

Qeru

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It is . The point is to take a DE that calls for partial fractions and get a solution without using that approach.

If it was then you are correct that would not be a common factor that could be separated. Also, we would need a definition of .
Yep thanks. I think thats where the confusion was for OP since the textbook makes it look like .
 

le_420_prince

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CM_Tutor, thankyou so much for the solution!! and thankyou everyone for the input also!!! a massive help!!
i think i was quite confused by the P(subscript)y which is now agreed to be a misprint and instead P*y but now that clears it all up
 

CM_Tutor

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At , we know that and so








Checks

When :



as expected.

Further, as :



Further, as :



We can see that the domain is and that is bounded below by 0 and bounded above by . That is, . It follows that, provided :



And thus is increasing at all times.

The answer to part (f) is thus:



and can also be expressed as:

 
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