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

In summary, the conversation is about a problem involving finding y as a function of x in an equation with multiple variables. The equation is not homogenous and does not mention being centered around a specific value. The participants discuss solving the problem using Euler equations and suggest using the substitution u = ln x to reduce it to a 2nd order linear equation with constant coefficients.
  • #1
mr_coffee
1,629
1
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 [Broken]
http://cwcsrv11.cwc.psu.edu/webwork2_files/tmp/equations/c9/0bc438e10b7ec37bb81942786780e41.png [Broken]
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! :biggrin:
 
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  • #2
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
 
  • #3
p.s. I'm new here. Where is everyone getting their maths type?
 
  • #4
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?
 
  • #5
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...
 
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  • #6
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?
 
  • #7
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 [Broken]
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 [Broken]
and got it right, but it was equated to 0, not x
 
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  • #8
mr_coffee said:
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 [Broken]
http://cwcsrv11.cwc.psu.edu/webwork2_files/tmp/equations/c9/0bc438e10b7ec37bb81942786780e41.png [Broken]
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! :biggrin:

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
 
Last edited by a moderator:
  • #9
topsquark, what are u subbing? u = ln x? where do u put that?
 
  • #10
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= eu so x3= e3u Your equation becomes
d2y/du2+ dy/du- 30u= e3u.
That should be easy.
 

1. What is a second order differential equation?

A second order differential equation is a mathematical equation that involves a function and its second derivative, or rate of change. It is commonly used to model physical systems and predict their behavior.

2. How can I determine which method to use for solving a second order differential equation?

The method you use will depend on the type of differential equation and its initial conditions. Some commonly used methods include separation of variables, substitution, and variation of parameters.

3. Can I use any method to solve a second order differential equation?

Not all methods are applicable to every type of second order differential equation. It is important to understand the characteristics of your equation and choose a method that is appropriate for it.

4. Do I need to have a strong background in mathematics to solve a second order differential equation?

A basic understanding of calculus and algebra is necessary to solve a second order differential equation. However, with practice and the use of resources such as textbooks and online tutorials, it is possible to learn the necessary skills.

5. Are there any common mistakes to avoid when solving a second order differential equation?

One common mistake is not checking the solution for extraneous solutions or missed solutions. It is also important to correctly apply any initial conditions and properly simplify your final answer.

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