There are 2 people
(unless they share books xD)
<u>Answer-</u>
![The \ integrating \ factor \ \mu (x) = e^{x^{2}}](https://tex.z-dn.net/?f=The%20%5C%20integrating%20%5C%20factor%20%5C%20%5Cmu%20%28x%29%20%3D%20e%5E%7Bx%5E%7B2%7D%7D)
<u>Solution-</u>
Given differential equation is,
![{y}'+2xy=8x](https://tex.z-dn.net/?f=%7By%7D%27%2B2xy%3D8x)
Comparing it with the general equation of first order linear equation, we get that,
![p(x)=2x \ and \ f(x)=8x](https://tex.z-dn.net/?f=p%28x%29%3D2x%20%5C%20and%20%5C%20f%28x%29%3D8x)
Now, calculating the value of the integrating factor,
![I.F. = \mu(x)=e^{\int p(x)dx} = e^{\int 2xdx} = e^{x^{2}}](https://tex.z-dn.net/?f=I.F.%20%3D%20%5Cmu%28x%29%3De%5E%7B%5Cint%20p%28x%29dx%7D%20%3D%20e%5E%7B%5Cint%202xdx%7D%20%3D%20e%5E%7Bx%5E%7B2%7D%7D)
♥ To solve this you are going to find x. ♥
♥ X in this case is representing a unknown variable, meaning we are not sure of the number x, and to finish the solution to your problem we have to find x.
♥ So to find x the first step is to simplify both sides of the equation.
♥ Starting with distribution:
![6=(-3)(x)+(-3)(2) \\ 6=-3x+-6 \\ 6=-3x-6](https://tex.z-dn.net/?f=6%3D%28-3%29%28x%29%2B%28-3%29%282%29%20%5C%5C%206%3D-3x%2B-6%20%5C%5C%206%3D-3x-6)
♥ Now we want to flip the equation.
![-3x-6=6](https://tex.z-dn.net/?f=-3x-6%3D6)
♥ Our next and almost final step, is to add the number -3 to both sides of our problem. This is going to help us tremendously.
![-3x-6+6=6+6 \\ -3x=12](https://tex.z-dn.net/?f=-3x-6%2B6%3D6%2B6%20%5C%5C%20-3x%3D12)
♥ Now our last step before we get our answer is to divide both sides of the problem. In this case we are going to do this:
![\frac{-3x}{-3} = \frac{12}{-3}](https://tex.z-dn.net/?f=%20%5Cfrac%7B-3x%7D%7B-3%7D%20%3D%20%20%5Cfrac%7B12%7D%7B-3%7D%20)
♥ And then because of that we get our final answer of: -4.
♥
x=-4 <span>♥</span>
It’s 27 I’m pretty sure
Because 18 divided by 2 gives you 9, then you would add the 9 onto the 18
The LCM<span> of 42 and 54 is 378.</span>