0,3 and 4,6
plug n check
3=0+3 is true
6=3+3 is also true
Factors of 15 are 1,3,5 and 15
Factors of 45 are 1,3,5,9,15,45
Factors of 90 are 1,2,3,5,6,9,10,15,18, 30,45,90
Now just check the numbers between those and you should have your answer
Step-by-step explanation:
We are given:
![xy'=coty](https://tex.z-dn.net/?f=xy%27%3Dcoty)
This can be rewritten as
![x\frac{dy}{dx} =coty](https://tex.z-dn.net/?f=x%5Cfrac%7Bdy%7D%7Bdx%7D%20%3Dcoty)
Next, we can bring the x's and y's to their respective sides by dividing by coty and x and then multiplying the dx to the other side. We can then change
into
. This gives us the differential
![tany\,dy=\frac{1}{x} \,dx](https://tex.z-dn.net/?f=tany%5C%2Cdy%3D%5Cfrac%7B1%7D%7Bx%7D%20%5C%2Cdx)
Now we can integrate each side
![\int tan(y)\,dy=\int \frac{1}{x} \,dx](https://tex.z-dn.net/?f=%5Cint%20tan%28y%29%5C%2Cdy%3D%5Cint%20%5Cfrac%7B1%7D%7Bx%7D%20%5C%2Cdx)
To integrate tan(y), we need to manipulate it
![\int \frac{sin(y)}{cos(y)} \,dy=\int \frac{1}{x} \,dx](https://tex.z-dn.net/?f=%5Cint%20%5Cfrac%7Bsin%28y%29%7D%7Bcos%28y%29%7D%20%5C%2Cdy%3D%5Cint%20%5Cfrac%7B1%7D%7Bx%7D%20%5C%2Cdx)
Now we can use u-substitution where ![u= cos(y)\\du=-sin(y) dy](https://tex.z-dn.net/?f=u%3D%20cos%28y%29%5C%5Cdu%3D-sin%28y%29%20dy)
This gives us
![-\int \frac{1}{u} \,du=\int \frac{1}{x} \,dx](https://tex.z-dn.net/?f=-%5Cint%20%5Cfrac%7B1%7D%7Bu%7D%20%5C%2Cdu%3D%5Cint%20%5Cfrac%7B1%7D%7Bx%7D%20%5C%2Cdx)
Now, lets integrate both sides
![-ln|u|=ln|x|+c](https://tex.z-dn.net/?f=-ln%7Cu%7C%3Dln%7Cx%7C%2Bc)
Next, we can substitute our u value back in
![-ln|cos(y)|=ln|x|+c](https://tex.z-dn.net/?f=-ln%7Ccos%28y%29%7C%3Dln%7Cx%7C%2Bc)
Now we can add
to the other side and subtract c from each side. This gives us
![C_2=ln|x|+ln|cos(y)|](https://tex.z-dn.net/?f=C_2%3Dln%7Cx%7C%2Bln%7Ccos%28y%29%7C)
Next, we can apply a property of logarithms to combine this sum of two logs into one log.
![C_2=ln|xcos(y)|](https://tex.z-dn.net/?f=C_2%3Dln%7Cxcos%28y%29%7C)
Lastly, we can add a base e to each side to remove the ln
![C_3=|xcos(y)|](https://tex.z-dn.net/?f=C_3%3D%7Cxcos%28y%29%7C)
And here is our answer.