Confused what method to use on this Diff EQ 2nd order

[mr_coffee]

ello ello!
I havn't seen an example of this type of problem. It has another variabler in the equation, i tried to divide by it, but its still in there. Here is the problem:
Find y as a function of x if
http://cwcsrv11.cwc.psu.edu/webwork2_files/tmp/equations/ee/448e92b4900790fd8de8a4207ce40d1.png
http://cwcsrv11.cwc.psu.edu/webwork2_files/tmp/equations/c9/0bc438e10b7ec37bb81942786780e41.png
y = ?
So i divided by x^2 and got:
y''+2y'/x -30y/x^2 = x;

This kind of looks like an integrating factor problem but not. ANy suggestions or can someone explain to me what's going on? Thanks!

 

[RooftopDuvet]

I've not looked at these type of equations before either but i'll speculate an answer if it's alright with the sire.

You're told the values of y' and y for when x is 1 so substitute them in with x as 1 to get:

(1^2)y'' + 2(1x5) - (30x6) = 1
y'' = 171

Then integrate to get: y' = 171x + c

but you're told that this is equal to 5 when x is 1 so that...

5 = 171 + c
c = -166
therefore y' = 171x - 166

then integrate again and find the constant again and you should get that..

y = 171(x^2)/2 -166x + 173/2

Don't hold me to that, I'm just frying logic

 

[RooftopDuvet]

p.s. I'm new here. Where is everyone getting their maths type?

 

[mr_coffee]

Rooftop, its LaTex.
And i subbmited that answer and it wasn't right, i have infinately many tries so I'm not worried about submitting wrong answers. Anyone else know?

 

[RooftopDuvet]

thanks for the heads up...

btw... you wouldn't happen to be doing some kind of radical calculus crossword puzzle would you?

edit: wait... i don't seem to be finding any LaTex. I may need a bigger heads up...

 

[HallsofIvy]

RooftopDuvet: y"= 171 only when x= 1. y'= 171x+ C only if y"= 171 for all x.

mr_coffee: You are asking the same kind of questions about the same kinds of problems over and over: This is an "Euler type" equation (also called "equipotential" equation). Remember what I told you about them last time?

 

[mr_coffee]

Sorry Ivey, i post too much! For it to be a Euler Type doesn't i have to "be centerd around" a value?
like http://tutorial.math.lamar.edu/AllBrowsers/3401/EulerEquations_files/eq0001M.gif
around Xo = 0. These type of differential equations are called Euler Equations. But mine doesn't loook like that because its not homogenous, and it doens't say its centered around anything, do i have to modify it or somthing?

I've solved Euler Equations beofre like this one:
http://cwcsrv11.cwc.psu.edu/webwork2_files/tmp/equations/26/a0c710e74525576a0593cd3a5bf7881.png
and got it right, but it was equated to 0, not x

 

[topsquark]

ello ello!
I havn't seen an example of this type of problem. It has another variabler in the equation, i tried to divide by it, but its still in there. Here is the problem:
Find y as a function of x if
http://cwcsrv11.cwc.psu.edu/webwork2_files/tmp/equations/ee/448e92b4900790fd8de8a4207ce40d1.png
http://cwcsrv11.cwc.psu.edu/webwork2_files/tmp/equations/c9/0bc438e10b7ec37bb81942786780e41.png
y = ?
So i divided by x^2 and got:
y''+2y'/x -30y/x^2 = x;

This kind of looks like an integrating factor problem but not. ANy suggestions or can someone explain to me what's going on? Thanks!

Worst comes to worst you can always use my favorite method: Use the substitution u = ln x. That reduces the problem to a 2nd order linear with constant coefficients.

-Dan

 

[mr_coffee]

topsquark, what are u subbing? u = ln x? where do u put that?

 

[HallsofIvy]

An "Euler type equation" has x to a power equal to the order of the derivative (that's why it's also called "equipotential"). Your example,
x²y"+ 2xy'- 30y= x³ is exactly of that kind. As for the substitution, u= ln x, I've also told you about that before.

If u= ln x, by the chain rule, dy/dx= (du/dx)(dy/du)= (1/x)dy/du and
d²y/dx²= (-1/x²)dy/du+ (1/x²)d²y/du². Also x= e^u so x³= e^3u Your equation becomes
d²y/du²+ dy/du- 30u= e^3u.
That should be easy.