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Toronto Math Forum
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MAT244--2019F
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MAT244--Lectures & Home Assignments
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Chapter 4
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4.2
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Topic: 4.2 (Read 1000 times)
jeyara85
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4.2
«
on:
November 13, 2019, 10:42:22 PM »
I was wondering if we will be expected to find the characteristic equations of higher-order equations (3rd,4th, ..., nth), or will we be provided the characteristic equation in the exam.
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david
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Re: 4.2
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Reply #1 on:
November 14, 2019, 04:09:16 PM »
Finding the characteristic equation for higher order equations is very similar to the second order case.
If we have the differential equation $a_ny^{(n)} + a_{n-1}y^{(n-1)} .... + a_1y' + a_0y = 0$ then the characteristic equation is $a_nr^n + a_{n-1}r^{n-1} .... + a_1r + a_0 = 0$.
To find the roots, we can use the fact that the product of the roots must be $a_0$ to help guess the roots.
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Last Edit: November 14, 2019, 04:13:43 PM by david
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Victor Ivrii
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Re: 4.2
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Reply #2 on:
November 15, 2019, 02:42:45 PM »
Yes, because for equations given they could be found easily
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ZeCheng Wu
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Re: 4.2
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Reply #3 on:
November 17, 2019, 10:25:51 PM »
what if a
_{(n)}
is not a constant, like x^2 for example
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david
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Re: 4.2
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Reply #4 on:
November 18, 2019, 09:36:48 AM »
If any of the $a_i$'s are not constant, then we cannot use the method above. Non-constant coefficient differential equations are generally harder to solve. We discussed a few methods in class such as reduction of order or using the Wronskian, but both methods require already knowing one solution.
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Toronto Math Forum
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MAT244--2019F
»
MAT244--Lectures & Home Assignments
»
Chapter 4
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4.2