Given the solution find the differential equation

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Homework Help Overview

The discussion revolves around identifying differential equations for given sets of functions, specifically exponential and polynomial functions. The original poster seeks to understand the process of deriving these equations as part of their preparation for a math physics course.

Discussion Character

  • Exploratory, Conceptual clarification, Problem interpretation

Approaches and Questions Raised

  • The original poster attempts to derive differential equations for the functions e-x, e-4x, and the polynomial set 1, x, x². They express uncertainty about their initial attempts and seek clarification on the correct approach.
  • Some participants question the original poster's understanding of characteristic equations and the relationship between the roots and the differential equations.
  • Others suggest focusing on the roots of the characteristic equations associated with the proposed solutions to guide the formulation of the differential equations.
  • There is a discussion about the need to combine solutions and the correct form of the equations, with some confusion regarding the notation and structure of differential equations.

Discussion Status

The discussion is ongoing, with participants providing feedback and corrections to the original poster's attempts. There is a recognition of the need for further clarification on the topic, and some guidance has been offered regarding the nature of the characteristic equations. However, there is no explicit consensus on the correct differential equations at this stage.

Contextual Notes

The original poster mentions having difficulty understanding the material due to past head trauma, which may affect their comprehension of the discussion. This context may influence the pace and depth of the conversation as they seek assistance.

JoshW
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Homework Statement


For each Set write the differential equation they are a solution to. It is also asked to find the wronskian but I remember how to do that.
(I am doing this on my own to brush up before I have to take mathphys next semester)
1. e-x,e-4x
2, 1,x,x2

Homework Equations

The Attempt at a Solution


1.(1+1/16)D"=e-x+e-4x
y"+y'+y=x2+x+1 but that doesn't seem rightIf someone could really explain how to do this or walk me through it I would be very happy!
 
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JoshW said:

Homework Statement


For each Set write the differential equation they are a solution to. It is also asked to find the wronskian but I remember how to do that.
(I am doing this on my own to brush up before I have to take mathphys next semester)
1. e-x,e-4x
2, 1,x,x2

Homework Equations

The Attempt at a Solution


1.(1+1/16)D"=e-x+e-4x
y"+y'+y=x2+x+1 but that doesn't seem right
Correct, it's not right. How did you get 1 + 1/16?

I'm hopeful that you still have your textbook from when you studied diff. equations. Some things are 1) the characteristic equation for a diff. equation, and 2) the Wronskian.

For the first problem, since you have two distinct exponentials, the diff. equation will be 2nd order, homogeneous, with constant coefficients.
For the second, since there are three solutions, the diff. equation will be 3rd order, homogeneous, with constant coefficients as well. The fact that the solutions are 1, x, and x2 is a clue to the number of distinct roots of the characteristic equation.
JoshW said:
If someone could really explain how to do this or walk me through it I would be very happy!
 
The book we used was Boas. So we may as well have not had a textbook at all.

So I need to find the roots to the equation? I am so lost.

so for the first my equation will be y"+y?

and the second I was correct? or for the second is it just a0x2+a1x+a2

I apologize I have trouble understanding what I read. I suffered head trauma as a kid and well I have trouble understanding anything unless it is spoken to me.
 
JoshW said:
The book we used was Boas. So we may as well have not had a textbook at all.

So I need to find the roots to the equation? I am so lost.
The roots of the characteristic equation. For example, the differential equation y' - 3y = 0 has a characteristic equation of r - 3 = 0.

The characteristic equation is obtained by assuming there is a solution of the form ##y = e^{rx}##
Differentiating, we get ## y' = re^{rx}##
Substituting into the diff. equation, we have
##y' - 3y = 0##, and ##y = e^{rx}##
##\Rightarrow re^{rx} - 3e^{rx} = 0##
##\Rightarrow (r - 3)e^{rx} = 0##
So r - 3 = 0 is the characteristic equation for the diff. equation y' - 3y = 0
The only root of this char. equation is r = 3, which gives us a solution of ##y = e^{3x}##
The general solution of y' - 3y = 0 is ##y = c_1e^{3x}##.

Another form of notation for the diff. equation uses the differentiation operator D. In this notation, the diff. equation would be (D - 3)y = 0. This notation is useful as you can pick off the characteristic equation directly.
JoshW said:
so for the first my equation will be y"+y?
No, for two reasons.
1. y'' + y is NOT an equation.
2. If the diff. eqn were y'' + y = 0, the characteristic equation would be ##r^2 + 1 = 0##.
JoshW said:
and the second I was correct?
No. At first I didn't think you had even done any work for #2 -- you didn't give any indication that your work was for problem 2.

JoshW said:
or for the second is it just a0x2+a1x+a2
No. The problem is to find a differential equation. The above is not an equation, nor is it anywhere close to what the differential equation would be whose solutions are 1, x, and x2.

You have a lot to catch up on. Let's focus on the first question before tackling the second one.
JoshW said:
I apologize I have trouble understanding what I read. I suffered head trauma as a kid and well I have trouble understanding anything unless it is spoken to me.
 
Last edited:
right ok. So for the first:
y=e-4x
y'=-4e-4x
y''=16e-4x

/∴ (y"-16y)=0 if y=e-4x
same thing for y=e-x

(y"-y)=0

I am not sure If I have to combine the solutions to form something elsefor #2
y'=2x
y"=2

/∴ (½y"+½y'+y)=0
 
JoshW said:
right ok. So for the first:
y=e-4x
y'=-4e-4x
y''=16e-4x

/∴ (y"-16y)=0 if y=e-4x
No.
For the equation you came up with, the fundamental solutions are ##y = e^{-4x}## and ##y = e^{4x}##
The characteristic equation of your diff. equation is ##r^2 - 16 = 0## or ##(r - 4)(r + 4) = 0##
You want a diff. equation whose characteristic equation has roots of -1 and -4, not 4 and -4.

JoshW said:
same thing for y=e-x

(y"-y)=0
Again, no.
Here the characteristic equation is ##r^2 - 1 = 0## or (r - 1)(r + 1) = 0, so the solutions are r = 1, or r = -1.
JoshW said:
I am not sure If I have to combine the solutions to form something elsefor #2
y'=2x
y"=2

/∴ (½y"+½y'+y)=0
No. Again, let's hold off on this one until you get squared away on the easier one, above.
 
How do I know that i want -1 and -4 as the roots?
 
JoshW said:
How do I know that i want -1 and -4 as the roots?
Because the two functions below are solutions to the diff. equation.
For each Set write the differential equation they are a solution to. It is also asked to find the wronskian but I remember how to do that.
...
1. e-x,e-4x
 
I appreciate the help! but this really isn't going anywhere. So thank you I will just wait until DE's are introduced in a class.
 

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