Separation of variables on 2nd order ode

In summary, the conversation discusses the difference between solving equations y' = x and y'' = x using separation of variables. It is also mentioned that y'' = 0 is equivalent to y''dx = 0dx = 0 and can be integrated to get y'(x) = C. The speaker also mentions that the derivative of a function being 0 means the function is a constant. The main question is then asked about why D^2y/Dx^2 = 0 is not the same as d^2y = 0 dx^2 and a rigorous explanation is requested.
  • #1
koab1mjr
107
0
Hi all

Quick one, if one had an equation y' = x on could simply separate the variables and integrate. Now it the equation y'' = x you would use separation of variables what drives this?

Also

y'' =0. Is the same as. y''dx =0 dx
Why is this legal?Thanks in advance
 
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  • #2
Well, you can always multiply both sides of an equation by the same thing! So, yes, y''= 0 is the same as y''dx= 0dx= 0. And you can now integrate both sides of the equation, with respect to x, to get y'(x)= C where C is a constant of integration. But you did not really need to do that. You know, I hope, that if the derivative of a function is 0, then the function is a constant: if the second derivative of y is 0, the first derivative is a constant.
 
  • #3
Ok re read what I wrote and my main question is not clear

Why is

D^2y/Dx^2 = 0 not the same as d^2y =0 dx^2

On one side u get a line on the other u get something else what is the rigorous explanation
 
Last edited:

1. What is separation of variables on 2nd order ODE?

Separation of variables is a method used to solve 2nd order ordinary differential equations (ODEs) by separating the variables into two separate equations and then solving each equation separately.

2. When is separation of variables applicable?

Separation of variables is applicable when the differential equation can be expressed as a product of two functions, each depending on only one variable.

3. What are the steps involved in separation of variables on 2nd order ODE?

The steps involved in separation of variables on 2nd order ODE are:
1. Rewrite the equation in its standard form.
2. Separate the variables by moving all terms containing one variable to one side and all terms containing the other variable to the other side.
3. Divide both sides by the function of one variable.
4. Integrate both sides with respect to the variable of that function.
5. Repeat the steps for the other variable.
6. Combine the two solutions to get the general solution.

4. What are the advantages of using separation of variables to solve 2nd order ODEs?

Some advantages of using separation of variables to solve 2nd order ODEs are:
1. It is a simple and systematic method.
2. It can be used to solve a wide range of 2nd order ODEs.
3. It allows for a general solution to be found.
4. It can be easily extended to higher order ODEs.

5. Are there any limitations to using separation of variables on 2nd order ODEs?

Yes, there are limitations to using separation of variables on 2nd order ODEs:
1. It can only be used for linear ODEs.
2. It cannot be used for nonlinear equations.
3. It may not always yield an explicit solution.
4. It may not be applicable to all initial value problems.

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