How Do You Expand a Differential Equation Solution into a Taylor Series?

[epheterson]

Many of you have probably used the book Differential Equations by Lomen & Lovelock.
For my class I'm working on Example 2, Page 153.

You don't need to see the book, though, to help me out. It's a four-part problem and I'm on the last step not knowing where to take it.

In Part B, we created a DE:

dP/dt = (397/4169) P + (6580236/4169)

and solved it:

P(t) = (397/8338) t^2 + (6580236/4169) t + (1225953/4169)

Now in Part D they ask:
"Expand the explicit solution you obtained in Part B in a Taylor Series about the origin."

I'm not so sure I've ever done anything like this, so I'm not clear on what steps I should take. I believe my teacher said I should find a similar Taylor Series and manipulate it to fit my needs but that still doesn't get me very far.

Absolutely any insight is appreciated, hope you can help me!

Thanks,
Eric

 

[HallsofIvy]

If your differential equation really is
dP/dt = (397/4169) P + (6580236/4169)
then dP/dt = (397/4169) P + (6580236/4169)
is NOT the solution. Your P(t) satisfies dP/dt = (397/4169) t + (6580236/4169).

The solution to dP/dt = (397/4169) P + (6580236/4169)
will involve an exponential function.

 

[epheterson]

I think I'm going to have to look at it in a different light. I could've sworn I put it correctly into maple, that's what it threw back at me.

Now that I try again, I get P(t) = -315680/19+exp((19/200)*t)*_C1, which I have checked by hand. I'm not entirely sure it's right though, I solved for C using a point from my original data table and got funny results when trying to use it.

I'll try again tomorrow. But now that I have an e in it, I believe I'll just be modifying the taylor series for e.

 

[epheterson]

Okay, I've figured out the problem. The one I posted was from a different part of the question.

I believe my correct equation is
$$P(t)=e^{.095t}16543 - 16571$$

Now I know that E is
$$e^{x} = \Sigma\frac{x^{n}}{n!}$$

So, correct me if I'm wrong, but do I manipulate the series as so to match my equation?

$$\Sigma[\frac{(0.095t)^n}{n!}16543-16571]$$

To be proper, all the summations here are from n=0 to $$\infty$$

In the question they ask me to relate it to the solution I got in part A, which is similar to what I originally posted:
$$P(t) = a t^2 + b t + c$$

I don't think how I have it set up results in an answer in that form. Did I turn it into a taylor series properly?

 

[HallsofIvy]

Okay, I've figured out the problem. The one I posted was from a different part of the question.

I believe my correct equation is
$$P(t)=e^{.095t}16543 - 16571$$

Now I know that E is
$$e^{x} = \Sigma\frac{x^{n}}{n!}$$

So, correct me if I'm wrong, but do I manipulate the series as so to match my equation?

$$\Sigma[\frac{(0.095t)^n}{n!}16543-16571]$$

No, the "-16571" is NOT part of the exponential and is NOT summed an infinite number of times!

To be proper, all the summations here are from n=0 to $$\infty$$

In the question they ask me to relate it to the solution I got in part A, which is similar to what I originally posted:
$$P(t) = a t^2 + b t + c$$

I don't think how I have it set up results in an answer in that form. Did I turn it into a taylor series properly?

And, again, that quadratic is WRONG.

 

[epheterson]

Well stay with me here...

I was unsure and that's what I was asking about, now as I understand it my new equation should be:
$$P(t) = (16543)\Sigma[\frac{(0.095t)^n}{n!}]-16571$$

True, the $$P(t) = a t^2 + b t + c$$ is incorrect for the original DE I gave you, but it is correct for a different part of the problem. This step is relating the quadratic to our new equation which we're making with summations.


I think I did this right, to relate them, I calculated out the summation to three terms and plugged them into get:
$$P(t) = 74.7t^2+1571.6t-28$$

which I can relate to my quadratic using similarities.

 

[HallsofIvy]

Well stay with me here...

I was unsure and that's what I was asking about, now as I understand it my new equation should be:
$$P(t) = (16543)\Sigma[\frac{(0.095t)^n}{n!}]-16571$$

Good! This is now correct.

True, the $$P(t) = a t^2 + b t + c$$ is incorrect for the original DE I gave you, but it is correct for a different part of the problem. This step is relating the quadratic to our new equation which we're making with summations.


I think I did this right, to relate them, I calculated out the summation to three terms and plugged them into get:
$$P(t) = 74.7t^2+1571.6t-28$$

which I can relate to my quadratic using similarities.

Okay, I didn't do the calculations myself but assuming the arithmetic is correct, this is the "2nd order Taylor Polynomial" for your function.