Answer:
p = 15/x
x= -3
Step-by-step explanation:
For the first problem, we can expand the equation to 4px+4=64
then simplify it to:
4px=60
then divide 4x from both sides of the equation
p=60/4x
then simplify:
p=15/x
For the second problem:
plug in -5 for p so the equation would look like
4(-5x +1)=64
simplify
-20x=60
x= -3
Using the given table of transforms,
![L\left\{ y'' - y' - 2y \right\} = L\left\{ 1-x \right\}](https://tex.z-dn.net/?f=L%5Cleft%5C%7B%20y%27%27%20-%20y%27%20-%202y%20%5Cright%5C%7D%20%3D%20L%5Cleft%5C%7B%201-x%20%5Cright%5C%7D)
![(s^2 Y(s) - sy(0) - y'(0)) - (s Y(s) - y(0)) - 2 Y(s) = \dfrac1s - \dfrac1{s^2}](https://tex.z-dn.net/?f=%28s%5E2%20Y%28s%29%20-%20sy%280%29%20-%20y%27%280%29%29%20-%20%28s%20Y%28s%29%20-%20y%280%29%29%20-%202%20Y%28s%29%20%3D%20%5Cdfrac1s%20-%20%5Cdfrac1%7Bs%5E2%7D)
(where Y(s) is the Laplace transform of y(x))
Solve for Y(s) :
![(s^2 - s - 2) Y(s) - s = \dfrac1s - \dfrac1{s^2}](https://tex.z-dn.net/?f=%28s%5E2%20-%20s%20-%202%29%20Y%28s%29%20-%20s%20%3D%20%5Cdfrac1s%20-%20%5Cdfrac1%7Bs%5E2%7D)
![(s^2 - s - 2) Y(s) - s = \dfrac{s-1}{s^2}](https://tex.z-dn.net/?f=%28s%5E2%20-%20s%20-%202%29%20Y%28s%29%20-%20s%20%3D%20%5Cdfrac%7Bs-1%7D%7Bs%5E2%7D)
![(s^2 - s - 2) Y(s) = \dfrac{s-1}{s^2} + s](https://tex.z-dn.net/?f=%28s%5E2%20-%20s%20-%202%29%20Y%28s%29%20%3D%20%5Cdfrac%7Bs-1%7D%7Bs%5E2%7D%20%2B%20s)
![(s^2 - s - 2) Y(s) = \dfrac{s^3 + s - 1}{s^2}](https://tex.z-dn.net/?f=%28s%5E2%20-%20s%20-%202%29%20Y%28s%29%20%3D%20%5Cdfrac%7Bs%5E3%20%2B%20s%20-%201%7D%7Bs%5E2%7D)
![Y(s) = \dfrac{s^3 + s - 1}{s^2 (s^2 - s - 2)}](https://tex.z-dn.net/?f=Y%28s%29%20%3D%20%5Cdfrac%7Bs%5E3%20%2B%20s%20-%201%7D%7Bs%5E2%20%28s%5E2%20-%20s%20-%202%29%7D)
![Y(s) = \dfrac{s^3 + s - 1}{s^2 (s + 1) (s - 2)}](https://tex.z-dn.net/?f=Y%28s%29%20%3D%20%5Cdfrac%7Bs%5E3%20%2B%20s%20-%201%7D%7Bs%5E2%20%28s%20%2B%201%29%20%28s%20-%202%29%7D)
Decompose the right side into partial fractions:
![Y(s) = \dfrac as + \dfrac b{s^2} + \dfrac c{s+1} + \dfrac d{s-2}](https://tex.z-dn.net/?f=Y%28s%29%20%3D%20%5Cdfrac%20as%20%2B%20%5Cdfrac%20b%7Bs%5E2%7D%20%2B%20%5Cdfrac%20c%7Bs%2B1%7D%20%2B%20%5Cdfrac%20d%7Bs-2%7D)
Solve for the coefficients.
![\implies s^3 + s - 1 = as(s+1)(s-2) + b(s+1)(s-2) + cs^2(s-2) + ds^2(s+1)](https://tex.z-dn.net/?f=%5Cimplies%20s%5E3%20%2B%20s%20-%201%20%3D%20as%28s%2B1%29%28s-2%29%20%2B%20b%28s%2B1%29%28s-2%29%20%2B%20cs%5E2%28s-2%29%20%2B%20ds%5E2%28s%2B1%29)
![\implies s^3 + s - 1 = -2 b + (-2 a - b) s + (-a + b - 2 c + d) s^2 + (a + c + d) s^3](https://tex.z-dn.net/?f=%5Cimplies%20s%5E3%20%2B%20s%20-%201%20%3D%20-2%20b%20%2B%20%28-2%20a%20-%20b%29%20s%20%2B%20%28-a%20%2B%20b%20-%202%20c%20%2B%20d%29%20s%5E2%20%2B%20%28a%20%2B%20c%20%2B%20d%29%20s%5E3)
![\implies \begin{cases}-2b = -1 \\ -2a-b = 1 \\ -a+b-2c+d = 0 \\ a+c+d = 1 \end{cases} \implies a=-\dfrac34, b=\dfrac12, c=1, d=\dfrac34](https://tex.z-dn.net/?f=%5Cimplies%20%5Cbegin%7Bcases%7D-2b%20%3D%20-1%20%5C%5C%20-2a-b%20%3D%201%20%5C%5C%20-a%2Bb-2c%2Bd%20%3D%200%20%5C%5C%20a%2Bc%2Bd%20%3D%201%20%5Cend%7Bcases%7D%20%5Cimplies%20a%3D-%5Cdfrac34%2C%20b%3D%5Cdfrac12%2C%20c%3D1%2C%20d%3D%5Cdfrac34)
Then
![Y(s) = -\dfrac34 \times \dfrac1s + \dfrac12 \times \dfrac1{s^2} + \dfrac1{s+1} + \dfrac34 \times\dfrac1{s-2}](https://tex.z-dn.net/?f=Y%28s%29%20%3D%20-%5Cdfrac34%20%5Ctimes%20%5Cdfrac1s%20%2B%20%5Cdfrac12%20%5Ctimes%20%5Cdfrac1%7Bs%5E2%7D%20%2B%20%5Cdfrac1%7Bs%2B1%7D%20%2B%20%5Cdfrac34%20%5Ctimes%5Cdfrac1%7Bs-2%7D)
Take the inverse transform and solve for y(x) :
![F(s) = \dfrac1s \implies f(x) = 1](https://tex.z-dn.net/?f=F%28s%29%20%3D%20%5Cdfrac1s%20%5Cimplies%20f%28x%29%20%3D%201)
![F(s) = \dfrac1{s^2} \implies f(x) = x](https://tex.z-dn.net/?f=F%28s%29%20%3D%20%5Cdfrac1%7Bs%5E2%7D%20%5Cimplies%20f%28x%29%20%3D%20x)
Using the frequency-shifting property,
![F(s) = \dfrac1{s+1} \implies f(x) = e^{-x}](https://tex.z-dn.net/?f=F%28s%29%20%3D%20%5Cdfrac1%7Bs%2B1%7D%20%5Cimplies%20f%28x%29%20%3D%20e%5E%7B-x%7D)
![F(s) = \dfrac1{s-2} \implies f(x) = e^{2x}](https://tex.z-dn.net/?f=F%28s%29%20%3D%20%5Cdfrac1%7Bs-2%7D%20%5Cimplies%20f%28x%29%20%3D%20e%5E%7B2x%7D)
So, the particular solution to the ODE is
![\boxed{y(x) = -\dfrac34 + \dfrac x2 + e^{-x} + \dfrac{3e^{2x}}4}](https://tex.z-dn.net/?f=%5Cboxed%7By%28x%29%20%3D%20-%5Cdfrac34%20%2B%20%5Cdfrac%20x2%20%2B%20e%5E%7B-x%7D%20%2B%20%5Cdfrac%7B3e%5E%7B2x%7D%7D4%7D)
Answer:
x=![\frac{y}{m} -\frac{p}{m} +q](https://tex.z-dn.net/?f=%5Cfrac%7By%7D%7Bm%7D%20-%5Cfrac%7Bp%7D%7Bm%7D%20%2Bq)
Step-by-step explanation:
Isisisisjssjsjsnsjsisisi This is 40
Answer:
the aswer could be 18 or 12
Step-by-step explanation: