(z^8)/(z^3) | Numerade

Name: Problem 16, Chapter 5: Elementary Algebra
Uploaded: 2020-08-27T23:42:02-08:00
Duration: 13 min 14 s
Channel: Bryan Lynn
Description: (z^8)/(z^3)

Key Concepts

Exponentiation Laws

Exponentiation laws are a set of rules that govern the operations performed on exponents. These laws help simplify expressions involving powers, making it easier to handle operations such as multiplication, division, and raising powers to powers.

Quotient of Powers Rule

The quotient of powers rule states that when dividing two exponential expressions with the same base, you subtract the exponent of the denominator from the exponent of the numerator. In mathematical terms, for any non-zero base a and exponents m and n, a^m divided by a^n equals a^(m-n).

Transcript

00:01 Okay, so here we have the non -homogeneous equation, the third derivative of z plus three times the second derivative minus four times z is equal to e to the 2x.

00:21 So let's take a look at the corresponding auxiliary equation, which would be r cubed plus 3r squared minus 4.

00:33 We set that equal to 0, and we get r is equal to 1, negative 2, and negative 2.

00:56 I guess that means our homogeneous solution is going to be some constant times e to the power of x plus c2 times e to the negative 2x plus c2 times e to the negative 2x plus c3 times x times e to the negative 2x.

01:28 And this x here is due to the multiplicity of the negative 2.

01:35 So this means our fundamental solution set is going to be e to the power of x, e to the negative 2x, and x times e to the negative 2x.

01:53 So we want a particular solution.

02:13 So the particular solution will be of the form, some constant times e to the x plus v2, e to the negative 2x, plus v3e to the negative 2x times x, where v1, v2, and v3, and v3 are all functions of x.

02:51 So to determine those, we take our matrix where the first row is each of the functions from our function set.

03:09 The second row is the derivative, first derivative of them.

03:24 So negative 2x, e to the negative 2x plus e to the negative 2x.

03:32 And then the third row is going to be the second derivative on that first one.

03:37 The first column is nice and easy.

03:44 And the last one here is 4xe to the negative 2x, minus 4e to the negative 2x.

04:00 Okay, so we want to find the determinant of that, and when we do, we will get 9e to the negative 3x.

04:21 And then we also want to get our sub -determinants here.

04:38 So if this is w, so we would consider w1 would be negative 1 squared times.

05:09 So we're considering the functions e to the negative 2x and xe to the negative 2x.

05:26 So we're just taking the first derivatives.

05:40 Of course you have them above.

05:41 So there's no like calculations here, just some copying.

05:47 Okay, but when we calculate this determinant, we end up with e to the negative 4x...

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