I'm going to guess that you meant to include parentheses somewhere, so that the ODE is supposed to be
![y'=\dfrac{y^2x}{y^3+x^3}+\dfrac yx](https://tex.z-dn.net/?f=y%27%3D%5Cdfrac%7By%5E2x%7D%7By%5E3%2Bx%5E3%7D%2B%5Cdfrac%20yx)
Then substitute
so that
. Then
![xv'+v=\dfrac{x^3v^2}{x^3v^3+x^3}+v](https://tex.z-dn.net/?f=xv%27%2Bv%3D%5Cdfrac%7Bx%5E3v%5E2%7D%7Bx%5E3v%5E3%2Bx%5E3%7D%2Bv)
![xv'=\dfrac{v^2}{v^3+1}](https://tex.z-dn.net/?f=xv%27%3D%5Cdfrac%7Bv%5E2%7D%7Bv%5E3%2B1%7D)
which is separable as
![\dfrac{v^3+1}{v^2}\,\mathrm dv=\dfrac{\mathrm dx}x](https://tex.z-dn.net/?f=%5Cdfrac%7Bv%5E3%2B1%7D%7Bv%5E2%7D%5C%2C%5Cmathrm%20dv%3D%5Cdfrac%7B%5Cmathrm%20dx%7Dx)
Integrate both sides: on the left,
![\displaystyle\int\frac{v^3+1}{v^2}\,\mathrm dv=\int\left(v+\frac1{v^2}\right)\,\mathrm dv=\dfrac12v^2-\dfrac1v](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint%5Cfrac%7Bv%5E3%2B1%7D%7Bv%5E2%7D%5C%2C%5Cmathrm%20dv%3D%5Cint%5Cleft%28v%2B%5Cfrac1%7Bv%5E2%7D%5Cright%29%5C%2C%5Cmathrm%20dv%3D%5Cdfrac12v%5E2-%5Cdfrac1v)
The other side is trivial. We end up with
![\dfrac12v^2-\dfrac1v=\ln|x|+C](https://tex.z-dn.net/?f=%5Cdfrac12v%5E2-%5Cdfrac1v%3D%5Cln%7Cx%7C%2BC)
Solve in terms of
:
![\boxed{\dfrac{y^2}{2x^2}-\dfrac xy=\ln|x|+C}](https://tex.z-dn.net/?f=%5Cboxed%7B%5Cdfrac%7By%5E2%7D%7B2x%5E2%7D-%5Cdfrac%20xy%3D%5Cln%7Cx%7C%2BC%7D)
Answer:
37.57 + 1.95x
Step-by-step explanation:
The cost of the shoes is a constant, and we don't know how many socks she bought, but we know that for every pair of socks she bought, they cost would be 1.95. Since we haven't been given the total cost of the purchase, meaning how much the total was, this will be an expression. We will multiply an x against the amount (1.95), which will calculate the price for how many socks were purchased. This means the answer will be 37.57 + 1.95x.
I will clarify a little bit more on why the other answers are incorrect. The first option is 37.57x + 1.95. This is incorrect because the question is not asking how many shoes were bought, but it is questioning how many socks were bought. Option number two states 37.57x + 1.95x. This means that the number of socks and shoes are both unknown, and the question does not state that. Also, another point to make would be that we could also add these variables together, and we do not want that. Option number three states 37.57 + 1.95 + x. X cannot be a separate variable, because by stating that, it means that there would be one more object that was bought that is unknown. I hope this helps you understand my explanation a bit more.
Have a great day, and best of luck for your math problem!
Answer:
Fourth option: x=1±√47
Step-by-step explanation:
This is a quadratic equation, then we must equal to zero: Subtracting both sides of the equation x², 5x, and 39:
![2x^{2}+3x-7-x^{2}-5x-39=x^{2}+5x+39-x^{2}-5x-39](https://tex.z-dn.net/?f=2x%5E%7B2%7D%2B3x-7-x%5E%7B2%7D-5x-39%3Dx%5E%7B2%7D%2B5x%2B39-x%5E%7B2%7D-5x-39)
Subtracting like terms:
![x^{2}-2x-46=0](https://tex.z-dn.net/?f=x%5E%7B2%7D-2x-46%3D0)
Applying the quadratic formula:
![ax^{2}+bx+c=0; a=1, b=-2, c=-46](https://tex.z-dn.net/?f=ax%5E%7B2%7D%2Bbx%2Bc%3D0%3B%20a%3D1%2C%20b%3D-2%2C%20c%3D-46)
![x=\frac{-b+-\sqrt{b^{2}-4ac}}{2a}\\ x=\frac{-(-2)+-\sqrt{(-2)^{2}-4(1)(-46)}}{2(1)}\\ x=\frac{2+-\sqrt{4+184}}{2}\\ x=\frac{2+-\sqrt{188}}{2}\\ x=\frac{2+-\sqrt{4(47)}}{2}\\ x=\frac{2+-\sqrt{4}\sqrt{47}}{2}\\ x=\frac{2+-2\sqrt{47}}{2}\\ x=\frac{2}{2}+- \frac{2\sqrt{47}}{2}\\ x=1+-\sqrt{47}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B-b%2B-%5Csqrt%7Bb%5E%7B2%7D-4ac%7D%7D%7B2a%7D%5C%5C%20x%3D%5Cfrac%7B-%28-2%29%2B-%5Csqrt%7B%28-2%29%5E%7B2%7D-4%281%29%28-46%29%7D%7D%7B2%281%29%7D%5C%5C%20x%3D%5Cfrac%7B2%2B-%5Csqrt%7B4%2B184%7D%7D%7B2%7D%5C%5C%20x%3D%5Cfrac%7B2%2B-%5Csqrt%7B188%7D%7D%7B2%7D%5C%5C%20x%3D%5Cfrac%7B2%2B-%5Csqrt%7B4%2847%29%7D%7D%7B2%7D%5C%5C%20x%3D%5Cfrac%7B2%2B-%5Csqrt%7B4%7D%5Csqrt%7B47%7D%7D%7B2%7D%5C%5C%20x%3D%5Cfrac%7B2%2B-2%5Csqrt%7B47%7D%7D%7B2%7D%5C%5C%20x%3D%5Cfrac%7B2%7D%7B2%7D%2B-%20%5Cfrac%7B2%5Csqrt%7B47%7D%7D%7B2%7D%5C%5C%20x%3D1%2B-%5Csqrt%7B47%7D)
<h3>
Answer: 15</h3>
=========================================================
Explanation:
n = number of new flash cards practiced
2n = number of old flash cards practiced
The 2n is due to the fact that the number of old flash cards is double that of the new ones (eg: 10 is twice that of 5).
The two expressions must add to 45.
new + old = total
n + 2n = 45
3n = 45
n = 45/3
n = 15
2n = 2*15 = 30
Noah practiced 15 new flash cards and 30 old flashcards which is a total of 15+30 = 45. This confirms our answer.
Also note that the ratio of new to old is 15:30 which reduces to 1:2, then we can multiply both sides by 5 to get 5:10. This effectively shows that 15:30 and 5:10 are the same equivalent ratio (just in different forms). This helps confirm our answer.
Answer:
104
Step-by-step explanation:
By subtracting the 158 angle from 180 you get 22 for the small angle next to it
Add the 22 and the 54 to get 76
then subtract the 76 from 180 to get 104
: )