x*y' + y = 8x
y' + y/x = 8 .... divide everything by x
dy/dx + y/x = 8
dy/dx + (1/x)*y = 8
We have something in the form
y' + P(x)*y = Q(x)
which is a first order ODE
The integrating factor is
Multiply both sides by the integrating factor (x) and we get the following:
dy/dx + (1/x)*y = 8
x*dy/dx + x*(1/x)*y = x*8
x*dy/dx + y = 8x
y + x*dy/dx = 8x
Note the left hand side is the result of using the product rule on xy. We technically didn't need the integrating factor since we already had the original equation in this format, but I wanted to use it anyway (since other ODE problems may not be as simple).
Since (xy)' turns into y + x*dy/dx, and vice versa, this means
y + x*dy/dx = 8x turns into (xy)' = 8x
Integrating both sides with respect to x leads to
xy = 4x^2 + C
y = (4x^2 + C)/x
y = (4x^2)/x + C/x
y = 4x + Cx^(-1)
where C is a constant. In this case, C = -5 leads to a solution
y = 4x - 5x^(-1)
you can check this answer by deriving both sides with respect to x
dy/dx = 4 + 5x^(-2)
Then plugging this along with y = 4x - 5x^(-1) into the ODE given, and you should find it satisfies that equation.
Answer:
These numbers are the Powers of Two
Beginning with term #1 = 2, the next term is always 2 times the PRECEDING term.
Second term is two squared, or 2 times 2, namely four.
Third term, multiply that four by two, giving eight, also known as two cubed.
Fourth term is twice as much, namely sixteen.
Just keep on doubling!
Step-by-step explanation:
Answer:
Horizontal translation of the parent graph
Step-by-step explanation:
<h2><u>Definitions</u>:</h2>
In the <u>vertex form</u> of a quadratic function, f(x) = a(x - h)² + k, where:
- (h, k) = vertex of the graph
- <em>a</em> = determines the width and direction of the graph's opening.
A <u>horizonal translation</u> to the parent graph is given by, y = f(x - h), where:
- <em>h</em> > 0 ⇒ Horizontal translation of <em>h</em> units to the right
- <em>h</em> < 0 ⇒ Horizontal translation of |<em>h </em>| units to the left
In the graph of g(x) = (x + 12)², the <u>vertex</u> occurs at point (-12, 0).
While the <u>vertex</u> of the parent graph, f(x) = x² occurs at point, (0, 0).
<h2><u>Answers</u>:</h2>
Since the vertex of g(x) occurs at point, (-12, 0), substituting the value of (<em>h</em>, <em>k </em>) into the vertex form will result into:
g(x) = a(x - h)² + k
g(x) = [x - (-12)]² + 0
g(x) = (x + 12)² + 0
g(x) = (x + 12)²
Therefore, the graph of g(x) = (x + 12)² represents the horizontal translation of the parent graph, f(x) = x², where the graph of g(x) is <em>horizontally</em> translated 12 units to the left.
If Taryn mowed 9 lawns in 7.5 hours and earned $112.50, then her hourly rate is:
$112.50 / 7.5 hrs = $15 per hour
If Alistair mowed 7 lawns in 5 hrs and earned $122.50, then her hourly rate is:
$122.50 / 5 hrs = $24.50 per hour
To earn an additional $735 each, they should work the following hours.
Taryn: $735 / $15 = 49 hours
Alistain: $735 / $24.50 = 30 hours