[Homework Help] Solving Seperable Differential Equations

  • Thread starter IareBaboon
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Hi, this is my first time posting a homework help problem, so please excuse any mistakes I happen to make :) I actually have two problems I'm stuck on. I have been trying to figure it out, and I'm sure it's a very simple problem, but I guess I'm just not that bright. Anyways, here they are.

1. The problem statement, all variables and given/known data

1) Find the particular solution of the differential equation for:


2) Find all values of k for which the function [tex]y=sin(kt)[/tex] satisfies the differential equation [tex]y''+16y=0[/tex]

2. Relevant equations

For the first question, you're told to use the following initial condition: [tex]y(0)=4[/tex]

3. The attempt at a solution

So for the first question, what I did was bring the x's to the right side and then got stuck at the integral part, which is this:


The left hand side is easy to integrate, however how would I go about integrating the right hand side? That's where I'm getting stuck at. A quick rundown would be very appreciated.

As for the second question, what I think I'm supposed to do is take the derivative of [tex]y=sin(kt)[/tex] twice, so that I have the second derivative of y, then substitute it into the original equation. Would that be right?

Thank you very much for any help. Anything would be appreciated :)
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I'd use a u-substitution to integrate the right side.
as for the second one, if I am understanding you right, that is the right approach
Alright. Thanks guys :) I think I can do the first question now. But the second question is a tad bit confusing. So let's say I take the first derivative of [tex]y=sin(kt)[/tex] which becomes [tex]y'=kcos(kt)[/tex] right? And then I take the second derivative of that, which is [tex]y''=-k^2sin(kt)[/tex] right?

After finding the second derivative, I plug in [tex]y=sin(kt)[/tex] and [tex]y''=-k^2sin(kt)[/tex] into [tex]y''+16y=0[/tex] which then makes it into [tex]-k^2sin(kt)+16sin(kt)=0[/tex]. What do I do at this point to find all values of k that would satisfy the differential equation [tex]y''+16y=0[/tex]?

Sorry if it's confusing to read. But thanks a lot for the help :)
Factor out the sin(kt) term on the left hand side and it comes down to finding the zeros of both sin(kt) and 16-k^2, which is easy.
Oh wow I can't believe I missed something so obvious. Lol thanks a lot for pointing that out to me. I got the right answer now :)
Really sorry for double posting, but I'm still having troubles with the first question. This is what I've done so far after moving the x's to the right side:



U-substitution for right side





That's what I'm getting. I know it's wrong, because the intitial condition given was [tex]y(0)=4[/tex] and when I plug in 0 into the x's, I'm not getting 4. What am I doing wrong? (Sorry if I'm bothering you guys by now. I just really need to understand this stuff as I have an exam soon...)
all your integration steps are missing the "+C"

use y(0)=4 to solve for "C", and add it to your equation!
Ah, right, forgot about the C. But I dont think that's what the question is asking for. Here's the question (My homework is posted on the web, :))


EDIT: Woops, just saw the last line of your last post just now. Let me do that and I'll get back to you.
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Okay, so I got the C which is 2.085145784. So what I get now is [tex]y=\sqrt{\frac{-11(x^2+1)^\frac{1}{2}}{-3}}+2.085145784[/tex] yet when I submit it online, the webworks says I am wrong. I really don't get what I'm doing wrong.

i dunno.
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lol its fine. thank you for the help though, much appreciated.
The C should be inside the square root because it comes in integration when y is still squared:

\int-6ydy=\int\frac{-11x}{\sqrt{x^2+1}}dx \Leftrightarrow 3\int d(y^2)= 11\int{d(\sqrt{x^2+1)} \Leftrightarrow y^2 =\frac{11}{3} \sqrt{x^2+1} + C

Now solve for C using the initial condition.
Ah, thanks to everyone! I got the answer now :)
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