Standard reduction of order procedure: suppose there is a second solution of the form
![y_2(x)=v(x)y_1(x)](https://tex.z-dn.net/?f=y_2%28x%29%3Dv%28x%29y_1%28x%29)
, which has derivatives
![y_2=vx^4](https://tex.z-dn.net/?f=y_2%3Dvx%5E4)
![{y_2}'=v'x^4+4vx^3](https://tex.z-dn.net/?f=%7By_2%7D%27%3Dv%27x%5E4%2B4vx%5E3)
![{y_2}''=v''x^4+8v'x^3+12vx^2](https://tex.z-dn.net/?f=%7By_2%7D%27%27%3Dv%27%27x%5E4%2B8v%27x%5E3%2B12vx%5E2)
Substitute these terms into the ODE:
![x^2(v''x^4+8v'x^3+12vx^2)-7x(v'x^4+4vx^3)+16vx^4=0](https://tex.z-dn.net/?f=x%5E2%28v%27%27x%5E4%2B8v%27x%5E3%2B12vx%5E2%29-7x%28v%27x%5E4%2B4vx%5E3%29%2B16vx%5E4%3D0)
![v''x^6+8v'x^5+12vx^4-7v'x^5-28vx^4+16vx^4=0](https://tex.z-dn.net/?f=v%27%27x%5E6%2B8v%27x%5E5%2B12vx%5E4-7v%27x%5E5-28vx%5E4%2B16vx%5E4%3D0)
![v''x^6+v'x^5=0](https://tex.z-dn.net/?f=v%27%27x%5E6%2Bv%27x%5E5%3D0)
and replacing
![v'=w](https://tex.z-dn.net/?f=v%27%3Dw)
, we have an ODE linear in
![w](https://tex.z-dn.net/?f=w)
:
![w'x^6+wx^5=0](https://tex.z-dn.net/?f=w%27x%5E6%2Bwx%5E5%3D0)
Divide both sides by
![x^5](https://tex.z-dn.net/?f=x%5E5)
, giving
![w'x+w=0](https://tex.z-dn.net/?f=w%27x%2Bw%3D0)
and noting that the left hand side is a derivative of a product, namely
![\dfrac{\mathrm d}{\mathrm dx}[wx]=0](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dx%7D%5Bwx%5D%3D0)
we can then integrate both sides to obtain
![wx=C_1](https://tex.z-dn.net/?f=wx%3DC_1)
![w=\dfrac{C_1}x](https://tex.z-dn.net/?f=w%3D%5Cdfrac%7BC_1%7Dx)
Solve for
![v](https://tex.z-dn.net/?f=v)
:
![v'=\dfrac{C_1}x](https://tex.z-dn.net/?f=v%27%3D%5Cdfrac%7BC_1%7Dx)
![v=C_1\ln|x|+C_2](https://tex.z-dn.net/?f=v%3DC_1%5Cln%7Cx%7C%2BC_2)
Now
![y=C_1x^4\ln|x|+C_2x^4](https://tex.z-dn.net/?f=y%3DC_1x%5E4%5Cln%7Cx%7C%2BC_2x%5E4)
where the second term is already accounted for by
![y_1](https://tex.z-dn.net/?f=y_1)
, which means
![y_2=x^4\ln x](https://tex.z-dn.net/?f=y_2%3Dx%5E4%5Cln%20x)
, and the above is the general solution for the ODE.
Answer:
There are 36 elements in the sample space.
Step-by-step explanation:
We are given the following in the question:
Digits:
1, 2, 3, 4, 5, 6
Counting principle:
- It helps us to evaluate total number of events for a probability outcome.
- It is done by multiplying the events to get the total number of outcomes.
We need to select two digit from the set of given digits.
For the first number we have 6 possibilities that is 1, 2, 3, 4, 5, 6.Since sampling is done with replacement foe the second digit again we have 6 possibilities that is 1, 2, 3, 4, 5, 6.
Thus, number of elements in the sample space is given by
![n = 6\times 6 = 36](https://tex.z-dn.net/?f=n%20%3D%206%5Ctimes%206%20%3D%2036)
There are 36 elements in the sample space.
Answer:
No, it is not a right triangle.
Answer:
$0.059
Step-by-step explanation:
To find the unit price per sticker, divide the price by the amount of stickers.
$2/34 = $0.059