You have, correctly,

.

You have the correct partial fraction decomposition:

.

It is true that

Note this is equal to

.

and you are correct to force

.

You conclude that the tops are the same? Why do you say that?

The bottoms aren’t the same… there is no good reason why the tops are the same (top means numerator and bottom denominator).

We are trying to get

so we can conclude the tops are equal:

.

But we can only conclude this if the bottoms are equal too.

You need to write

as a single fraction with the same bottom as

Then you can force them equal to each other and conclude that

and find .

Regards,

J.P.

Where you have “redo but change missed a step” you do have an issue. So you have

.

This is not equal to

.

You have four options.

1. Divide both sides by to get

.

Now the issue is your fraction in your fraction. This is not good. To get rid of it you need to multiply above and below by :

.

2. Divide each of the three terms by separately:

.

You can send these all back separately or…

3. Write this as a single fraction:

4. The other thing you might have done was write the right hand side as a single fraction here:

.

Regards,

J.P.

It is first year engineering material rather than third year material.

You have to show that satisfies the differential equation.

So you have to calculate

(*)

and see do you get zero.

To calculate the second derivative you differentiate twice… so you start by differentiating once.

The thing in green is not finished, it hasn’t used the Chain Rule.

The chain rule says if you have a function of a function, (where ‘outside’ and ‘inside’), like here is the outside and is the inside, that the derivative is given by:

.

That is you differentiate the outside, evaluate at the inside, and then multiply by the derivative of what is inside.

With , fixing the constant

.

That is, after fixing the constant , the derivative of , which is , evaluated at the inside (so this instead of ), multiplied by the derivative of what is inside.

The last day you were tired and stuck on

.

This is just like differentiating say … the derivative of a constant times is just the constant; for a constant :

.

Therefore

,

so

.

Now you need to differentiate a second time to find the second derivative… then see does (*) give you zero.

Regards,

J.P.

Still stuck on the same question from the other day. It’s the only question I have left and I’m so confused. Question 3.5.1 c ii.

Is there an example of this question I can look at?

Thanks.

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