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*Elementary Differential Equations and Boundary Value Problems, 9th Edition*. The problem comes from Section 2.8: The Existence and Uniqueness Theorem and is part of a collection of problems intended to show that the sequence ##{\phi_n(t)}## converges, where

$$\phi_{n+1}(t) = \int_0^t f[s,\phi_n(s)] ds$$

with ##\phi_0(t) = 0##. The problem requires the following two results:

1) If ##\phi_{n-1}(t)## and ##\phi_n(t)## are members of the sequence ##{\phi_n(t)}## then

$$|f[t,\phi_n(t)]-f[t,\phi_{n-1}(t)| \leq K|\phi_n(t) - \phi_{n-1}(t)|$$

where ##K## is chosen to be the maximum value of ##\frac{∂f}{∂y}## in the region ##D##.

2) If ##|t|<h##, then

$$|\phi_1(t)| \leq M|t|$$

where ##M## is chosen so that ##|f(t,y)|\leq M## for ##(t,y)## in ##D##.

The problem is as follows: "Use the results of Problem 16 and part (a) of Problem 17 [Results 1 and 2 above, respectively] to show that ##|\phi_2(t) - \phi_1(t)| \leq \frac{MK|t|^2}{2}##."

(The problem and results are found in Boyce & DiPrima's

*Elementary Differential Equations and Boundary Value Problems, 9th Edition; Problem 2.8.17(b); pg. 120.)*

I've been struggling with this problem even though it

*seems*like it should be fairly straightforward. From now on I'm going to write ##\phi_n(t) = \phi_n## for the sake of laziness. I've been playing around with a few different approaches - I tried writing out the left-hand side of my desired result explicitly using the definition of ##\phi_n## which gives me

$$|\phi_2 - \phi_1| = | \int_0^t f(s,\phi_1) ds - \int_0^t f(s,\phi_0) ds | = | \int_0^t f(s,\phi_1) - f(s,\phi_0) ds | $$

I got stuck here. I can

*almost*apply Result 1 now, but not quite since I have my integrand inside the absolute value...and even if I could it didn't yield promising results. Then I thought of using the inequality ##|x-y| \leq | |x| - |y| |## which let me express ##|\phi_1| ## using Result 2 but I was again stuck with no way to write ##|\phi_2|##.

I noticed that the right-hand side of the first result looks very similar to the left-hand side of what I wish to show when you take ##n=2## but that didn't get me very far either:

$$|f[t,\phi_n(t)]-f[t,\phi_{n-1}(t)| \leq K|\phi_n(t) - \phi_{n-1}(t)|$$

So I could try to show

$$|f[t,\phi_n(t)]-f[t,\phi_{n-1}(t)| \leq \frac{MK^2|t|^2}{2}$$

But then I realized that this is only true if I assume that what I want to show is true and so this shouldn't be the approach I take.

Any guidance would be appreciated. I feel like this should be pretty straightforward, but I'm lost for ideas right now. Thanks in advance for any help!