The Formula Of Volume Is :
V = L*W*H. So, Lets Plug In The Known Values.
V = 16*16*16.
16*16 = 256.
256 * 16 = 4096 yr^3. This Is The Volume. Any Further Questions?
Answer:
<u>12n + 2</u>
Step-by-step explanation:
![w(n) = un + 2](https://tex.z-dn.net/?f=w%28n%29%20%3D%20un%20%2B%202)
![w(3n) = 4(3n) + 2](https://tex.z-dn.net/?f=w%283n%29%20%3D%204%283n%29%20%2B%202)
![ans = 12n + 2](https://tex.z-dn.net/?f=ans%20%3D%2012n%20%2B%202)
hope it helps you
have a great day!!
I'll assume the ODE is
![y'' - 3y' + 2y = e^x + e^{2x} + e^{-x}](https://tex.z-dn.net/?f=y%27%27%20-%203y%27%20%2B%202y%20%3D%20e%5Ex%20%2B%20e%5E%7B2x%7D%20%2B%20e%5E%7B-x%7D)
Solve the homogeneous ODE,
![y'' - 3y' + 2y = 0](https://tex.z-dn.net/?f=y%27%27%20-%203y%27%20%2B%202y%20%3D%200)
The characteristic equation
![r^2 - 3r + 2 = (r - 1) (r - 2) = 0](https://tex.z-dn.net/?f=r%5E2%20-%203r%20%2B%202%20%3D%20%28r%20-%201%29%20%28r%20-%202%29%20%3D%200)
has roots at
and
. Then the characteristic solution is
![y = C_1 e^x + C_2 e^{2x}](https://tex.z-dn.net/?f=y%20%3D%20C_1%20e%5Ex%20%2B%20C_2%20e%5E%7B2x%7D)
For nonhomogeneous ODE (1),
![y'' - 3y' + 2y = e^x](https://tex.z-dn.net/?f=y%27%27%20-%203y%27%20%2B%202y%20%3D%20e%5Ex)
consider the ansatz particular solution
![y = axe^x \implies y' = a(x+1) e^x \implies y'' = a(x+2) e^x](https://tex.z-dn.net/?f=y%20%3D%20axe%5Ex%20%5Cimplies%20y%27%20%3D%20a%28x%2B1%29%20e%5Ex%20%5Cimplies%20y%27%27%20%3D%20a%28x%2B2%29%20e%5Ex)
Substituting this into (1) gives
![a(x+2) e^x - 3 a (x+1) e^x + 2ax e^x = e^x \implies a = -1](https://tex.z-dn.net/?f=a%28x%2B2%29%20e%5Ex%20-%203%20a%20%28x%2B1%29%20e%5Ex%20%2B%202ax%20e%5Ex%20%3D%20e%5Ex%20%5Cimplies%20a%20%3D%20-1)
For the nonhomogeneous ODE (2),
![y'' - 3y' + 2y = e^{2x}](https://tex.z-dn.net/?f=y%27%27%20-%203y%27%20%2B%202y%20%3D%20e%5E%7B2x%7D)
take the ansatz
![y = bxe^{2x} \implies y' = b(2x+1) e^{2x} \implies y'' = b(4x+4) e^{2x}](https://tex.z-dn.net/?f=y%20%3D%20bxe%5E%7B2x%7D%20%5Cimplies%20y%27%20%3D%20b%282x%2B1%29%20e%5E%7B2x%7D%20%5Cimplies%20y%27%27%20%3D%20b%284x%2B4%29%20e%5E%7B2x%7D)
Substitute (2) into the ODE to get
![b(4x+4) e^{2x} - 3b(2x+1)e^{2x} + 2bxe^{2x} = e^{2x} \implies b=1](https://tex.z-dn.net/?f=b%284x%2B4%29%20e%5E%7B2x%7D%20-%203b%282x%2B1%29e%5E%7B2x%7D%20%2B%202bxe%5E%7B2x%7D%20%3D%20e%5E%7B2x%7D%20%5Cimplies%20b%3D1)
Lastly, for the nonhomogeneous ODE (3)
![y'' - 3y' + 2y = e^{-x}](https://tex.z-dn.net/?f=y%27%27%20-%203y%27%20%2B%202y%20%3D%20e%5E%7B-x%7D)
take the ansatz
![y = ce^{-x} \implies y' = -ce^{-x} \implies y'' = ce^{-x}](https://tex.z-dn.net/?f=y%20%3D%20ce%5E%7B-x%7D%20%5Cimplies%20y%27%20%3D%20-ce%5E%7B-x%7D%20%5Cimplies%20y%27%27%20%3D%20ce%5E%7B-x%7D)
and solve for
.
![ce^{-x} + 3ce^{-x} + 2ce^{-x} = e^{-x} \implies c = \dfrac16](https://tex.z-dn.net/?f=ce%5E%7B-x%7D%20%2B%203ce%5E%7B-x%7D%20%2B%202ce%5E%7B-x%7D%20%3D%20e%5E%7B-x%7D%20%5Cimplies%20c%20%3D%20%5Cdfrac16)
Then the general solution to the ODE is
![\boxed{y = C_1 e^x + C_2 e^{2x} - xe^x + xe^{2x} + \dfrac16 e^{-x}}](https://tex.z-dn.net/?f=%5Cboxed%7By%20%3D%20C_1%20e%5Ex%20%2B%20C_2%20e%5E%7B2x%7D%20-%20xe%5Ex%20%2B%20xe%5E%7B2x%7D%20%2B%20%5Cdfrac16%20e%5E%7B-x%7D%7D)
Answer:
![x = \frac{d+c}{a - b}](https://tex.z-dn.net/?f=x%20%3D%20%5Cfrac%7Bd%2Bc%7D%7Ba%20-%20b%7D)
Step-by-step explanation:
Equation: ![ax – c = bx + d](https://tex.z-dn.net/?f=ax%20%E2%80%93%20c%20%3D%20bx%20%2B%20d)
Jalil says it is not possible to isolate x because x has a different unknown coefficient.
Victoria t. Victoria believes there is a solution
Solving the equation:
![ax-c = bx + d](https://tex.z-dn.net/?f=ax-c%20%3D%20bx%20%2B%20d)
![ax - bx= d+c](https://tex.z-dn.net/?f=ax%20-%20bx%3D%20d%2Bc)
![(a - b)x = d+c](https://tex.z-dn.net/?f=%28a%20-%20b%29x%20%3D%20d%2Bc)
![x = \frac{d+c}{a - b}](https://tex.z-dn.net/?f=x%20%3D%20%5Cfrac%7Bd%2Bc%7D%7Ba%20-%20b%7D)
So, this shows x can be isolated .
Victoria was right .
It was not possible to isolate x if the coefficients of x would be same .
But in the given equation the coefficients of x are not same .
So, Victoria is right.
Answer:
Options A, C and D are correct ones.
Step-by-step explanation:
Options A, C and D are correct.