Answer: X would equal 7 so X=7
Suppose
is another solution. Then
![\begin{cases}y_2=vx^3\\{y_2}'=v'x^3+3vx^2//{y_2}''=v''x^3+6v'x^2+6vx\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7Dy_2%3Dvx%5E3%5C%5C%7By_2%7D%27%3Dv%27x%5E3%2B3vx%5E2%2F%2F%7By_2%7D%27%27%3Dv%27%27x%5E3%2B6v%27x%5E2%2B6vx%5Cend%7Bcases%7D)
Substituting these derivatives into the ODE gives
![x^2(v''x^3+6v'x^2+6vx)-x(v'x^3+3vx^2)-3vx^3=0](https://tex.z-dn.net/?f=x%5E2%28v%27%27x%5E3%2B6v%27x%5E2%2B6vx%29-x%28v%27x%5E3%2B3vx%5E2%29-3vx%5E3%3D0)
![x^5v''+5x^4v'=0](https://tex.z-dn.net/?f=x%5E5v%27%27%2B5x%5E4v%27%3D0)
Let
, so that
![\begin{cases}u=v'\\u'=v''\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7Du%3Dv%27%5C%5Cu%27%3Dv%27%27%5Cend%7Bcases%7D)
Then the ODE becomes
![x^5u'+5x^4u=0](https://tex.z-dn.net/?f=x%5E5u%27%2B5x%5E4u%3D0)
and we can condense the left hand side as a derivative of a product,
![\dfrac{\mathrm d}{\mathrm dx}[x^5u]=0](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dx%7D%5Bx%5E5u%5D%3D0)
Integrate both sides with respect to
:
![\displaystyle\int\frac{\mathrm d}{\mathrm dx}[x^5u]\,\mathrm dx=C](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint%5Cfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dx%7D%5Bx%5E5u%5D%5C%2C%5Cmathrm%20dx%3DC)
![x^5u=C\implies u=Cx^{-5}](https://tex.z-dn.net/?f=x%5E5u%3DC%5Cimplies%20u%3DCx%5E%7B-5%7D)
Solve for
:
![v'=Cx^{-5}\implies v=-\dfrac{C_1}4x^{-4}+C_2](https://tex.z-dn.net/?f=v%27%3DCx%5E%7B-5%7D%5Cimplies%20v%3D-%5Cdfrac%7BC_1%7D4x%5E%7B-4%7D%2BC_2)
Solve for
:
![\dfrac{y_2}{x^3}=-\dfrac{C_1}4x^{-4}+C_2\implies y_2=C_2x^3-\dfrac{C_1}{4x}](https://tex.z-dn.net/?f=%5Cdfrac%7By_2%7D%7Bx%5E3%7D%3D-%5Cdfrac%7BC_1%7D4x%5E%7B-4%7D%2BC_2%5Cimplies%20y_2%3DC_2x%5E3-%5Cdfrac%7BC_1%7D%7B4x%7D)
So another linearly independent solution is
.
to get his wage for the next day, you would do 18x1.2, which equals 21.6, so he would now be making $21.6 per hour.
Answer:
I agree with none of them
Step-by-step explanation:
Given
![Expression= 2^{28}](https://tex.z-dn.net/?f=Expression%3D%202%5E%7B28%7D)
![Andre: 2 * 28 = 56](https://tex.z-dn.net/?f=Andre%3A%202%20%2A%2028%20%3D%2056)
![Elena: 28 * 28 = 756](https://tex.z-dn.net/?f=Elena%3A%2028%20%2A%2028%20%3D%20756)
Required
Who is correct?
![Expression= 2^{28}](https://tex.z-dn.net/?f=Expression%3D%202%5E%7B28%7D)
When we apply law of indices to the above, it gives 2 multiplies by itself in 28 places
i.e.
![Expression= 2^{28}](https://tex.z-dn.net/?f=Expression%3D%202%5E%7B28%7D)
---- in 28 places
Andre's interpretation of the expression is incorrect as:
![2^{28} \neq 56](https://tex.z-dn.net/?f=2%5E%7B28%7D%20%5Cneq%2056)
Also, Elena's expression is incorrect.
This is so, because
![2^28 \neq 28*28](https://tex.z-dn.net/?f=2%5E28%20%5Cneq%2028%2A28)
<em>Hence, both of them are incorrect and I agree with none of them</em>