Answer:
Machine 1: 165
Machine 2: 210
Step-by-step explanation:
Machine 1: 11 bottle per minute
11 * 15 = 165 labeled bottles per minute
Machine 2: 14 bottles per minute
14*15 = 210 labeled bottles per minute
Both equations are linear, so I'll use the integrating factor method.
The first ODE

has integrating factor

In the original equation, multiply both sides by eˣ :

Observe that
d/dx [xeˣ] = eˣ + xeˣ = (x + 1) eˣ
so that the left side is the derivative of a product, namely

Integrate both sides with respect to x :


Solve for y :

Use the given initial condition to solve for C. When x = 1, y = 2, so

Then the particular solution is

The second ODE

has integrating factor

Multiply both sides of the equation by 1/(1 + x²) :

and observe that
d/dx[1/(1 + x²)] = -2x/(1 + x²)²
Then



When x = 0, y = 3, so


Answer:
1) 5x(1 - 3x)
2) 3(x - 3)
Step-by-step explanation:
<u>Question # 1:</u>
= 5x - 15x²
Take 5x common
= 5x(1 - 3x)
<u>Question # 2:</u>
= 3x - 9
Take 3 common
= 3(x - 3)
![\rule[225]{225}{2}](https://tex.z-dn.net/?f=%5Crule%5B225%5D%7B225%7D%7B2%7D)
Hope this helped!
<h3>~AH1807</h3>
Answer:
Assuming you mean the square root of 23, it is irrational.
I believe the slope is 0 because it they are both horizontal lines? I may be wrong though.