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July 7th, 2011, 10:32 AM   #1
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Differential equation problem

[attachment=0:3arcjy5l]untitled_1.png[/attachment:3arcjy5l]

I have the following problem from a book I'm reading, it occurs several chapters before integration and DE's are covered!

The current in the RCL-circuit above must satisfy the equation



If where and are constants, show that a solution is



where and are any constants whatsoever and

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July 7th, 2011, 01:53 PM   #2
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Re: Differential equation problem

Perhaps they intend for you to take the given solution, find its first and second derivatives with respect to t, then substitute into the ODE to show that it satisfies the equation. However, here is how I would derive the solution from the given information:

Given then and the ODE is then:



First, we find the solution to the corresponding homogeneous equation, whose associated auxiliary equation is:



Application of the quadratic formula yields:



With we have

Since the auxiliary equation has complex conjugate roots, this means:

where A and B are constants.

Now we may use the method of undetermined coefficients to find the particular solution . We begin by assuming it will be of the form:



where are the coefficients to be determined. Differentiating with respect to t, we find:





Substituting into the original ODE we have:







Equating coefficients across the equation yields the system:





Solving the 2nd equation for we find:



Substituting for into the 1st equation gives:



Solving for gives:



And hence:



Thus, our particular solution is:





Using the linear combination identity we may write this as:



where

Defining we may now write:



Now, the general solution f(t) is found by the superposition:

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July 7th, 2011, 05:30 PM   #3
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Re: Differential equation problem

Here are a few words on some of the methods I used. First, consider the following homogeneous equation:



Because of the nature of the equation, it is reasonable to assume a solution will be of the form:

where

Differentiating with respect to x, we find:





Substituting into the ODE gives:





Since we are left with the auxiliary or characteristic equation:



Thus, by the quadratic formula, we find the roots of the auxiliary equation to be:



If we define:





Then we have two solutions:





where are arbitrary constants. Thus, we know:





Adding the two equations, we get:



Now, if we define:

then



and we may now write:



where

Now, if the roots of the auxiliary equation are complex conjugates, then we may express them as:





where:





Thus, we may express the solution as:



Now, using Euler's formula we may express the solution as:



Since and we may write:





Since are arbitrary constants, we may write:



Now, as we saw above, we know:





Are both linearly independent solutions to the ODE, thus if we define:

then we have:





Since is a solution, then so is and we may express the general solution as:



Now, consider the inhomogeneous equation:



and the corresponding homogeneous equation:



Suppose is a particular solution to the inhomogeneous equation and is the solution to the homogeneous equation. Note that the two solutions must be linearly independent, we then have:





Adding the two equations, we get:



With we now have:



The particular solution will depend on the form taken by g(x) naturally, and your book should have a table listing the form it will take.
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July 7th, 2011, 05:35 PM   #4
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Re: Differential equation problem

Thanks!
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July 7th, 2011, 06:26 PM   #5
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What is represented by the little yellow blob on the left-hand side of the diagram?
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July 7th, 2011, 06:35 PM   #6
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Re: Differential equation problem

I don't see it. I don't think it means anything, whatever it is, the diagram is just a resistor, inductor and capacitor with some other component E, connected by lines. There is some discoloration on the lines.
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July 7th, 2011, 07:47 PM   #7
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I assume the gaps in the wiring lines are similarly unintended. The "other component" you refer to is presumably a power supply.
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July 7th, 2011, 09:03 PM   #8
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Nice solution, MarkFL!
More info on electromagnetism is here.
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July 7th, 2011, 09:10 PM   #9
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Re: Differential equation problem

Thanks!

It's interesting that an under-damped spring-mass system subject to an external sinusoidal force is governed by exactly the same type of ODE.

Note that as , we have thus this is called the transient solution, and the other term is called the steady-state solution.
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July 8th, 2011, 07:15 AM   #10
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Re:

Quote:
Originally Posted by skipjack
I assume the gaps in the wiring lines are similarly unintended.
Yes.

Quote:
Originally Posted by skipjack
The "other component" you refer to is presumably a power supply.
I don't know. It oscillates.
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