<h2><u>Circle Equations</u></h2>
<h3>Write the standard form of the equation of the circle with the given characteristics.</h3><h3>Center: (0, 0); Radius: 2</h3>
To determine the equation of a circle, use the standard form of a circle (x - h)² + (y - k)² = r² where,
- <u>(h, k)</u> is the center; and
- <u>r</u> is the radius
Substitute the values of the center and radius to the standard form.
<u>Given:</u>
<u>(0, 0)</u> - <u>center</u>
<u>2</u> - <u>radius</u>
- (x - h)² + (y - k)² = 2²
- (x - 0)² + (y - 0)² = 4
- x² + y² = 4
<u>Answer:</u>
- The equation of the circle is <u>x² + y² = 4</u>.
Wxndy~~
Answer:
Geometric Sequence
Step-by-step explanation:
1. Check the difference.
The difference between the 1st and 2nd term

The difference between the 2nd and 3rd term

The difference is not the same. Therefore, it is not an arithmetic sequence.
2. Check the ratio
The ratio between the 1st and 2nd term

The ratio between the 2nd and 3rd term

The ratio is the same. Therefore, it is a geometric sequence.
F(-1) = 12 - 5(-1)
f(-1) = 12 + 5
Solution: f(-1) = 17
Take the homogeneous part and find the roots to the characteristic equation:

This means the characteristic solution is

.
Since the characteristic solution already contains both functions on the RHS of the ODE, you could try finding a solution via the method of undetermined coefficients of the form

. Finding the second derivative involves quite a few applications of the product rule, so I'll resort to a different method via variation of parameters.
With

and

, you're looking for a particular solution of the form

. The functions

satisfy


where

is the Wronskian determinant of the two characteristic solutions.

So you have




So you end up with a solution

but since

is already accounted for in the characteristic solution, the particular solution is then

so that the general solution is