Given the solution find the differential equation

[JoshW]

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!

 

[Mark44]

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.

If someone could really explain how to do this or walk me through it I would be very happy!

 

[JoshW]

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.

 

[Mark44]

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.

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##.

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.

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.

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]

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

 

[Mark44]

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.

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.

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.

 

[JoshW]

How do I know that i want -1 and -4 as the roots?

 

[Mark44]

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

 

[JoshW]

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.