This DE has characteristic equation
![4r^2 - 12r + 9r = (2r - 3)^2 = 0](https://tex.z-dn.net/?f=4r%5E2%20-%2012r%20%2B%209r%20%3D%20%282r%20-%203%29%5E2%20%3D%200)
with a repeated root at r = 3/2. Then the characteristic solution is
![y_c = C_1 e^{\frac32 x} + C_2 x e^{\frac32 x}](https://tex.z-dn.net/?f=y_c%20%3D%20C_1%20e%5E%7B%5Cfrac32%20x%7D%20%2B%20C_2%20x%20e%5E%7B%5Cfrac32%20x%7D)
which has derivative
![{y_c}' = \dfrac{3C_1}2 e^{\frac32 x} + \dfrac{3C_2}2 x e^{\frac32x} + C_2 e^{\frac32 x}](https://tex.z-dn.net/?f=%7By_c%7D%27%20%3D%20%5Cdfrac%7B3C_1%7D2%20e%5E%7B%5Cfrac32%20x%7D%20%2B%20%5Cdfrac%7B3C_2%7D2%20x%20e%5E%7B%5Cfrac32x%7D%20%2B%20C_2%20e%5E%7B%5Cfrac32%20x%7D)
Use the given initial conditions to solve for the constants:
![y(0) = 3 \implies 3 = C_1](https://tex.z-dn.net/?f=y%280%29%20%3D%203%20%5Cimplies%203%20%3D%20C_1)
![y'(0) = \dfrac52 \implies \dfrac52 = \dfrac{3C_1}2 + C_2 \implies C_2 = -2](https://tex.z-dn.net/?f=y%27%280%29%20%3D%20%5Cdfrac52%20%5Cimplies%20%5Cdfrac52%20%3D%20%5Cdfrac%7B3C_1%7D2%20%2B%20C_2%20%5Cimplies%20C_2%20%3D%20-2)
and so the particular solution to the IVP is
![\boxed{y(x) = 3 e^{\frac32 x} - 2 x e^{\frac32 x}}](https://tex.z-dn.net/?f=%5Cboxed%7By%28x%29%20%3D%203%20e%5E%7B%5Cfrac32%20x%7D%20-%202%20x%20e%5E%7B%5Cfrac32%20x%7D%7D)
12:45Answer:
Part A: 9 hours 45 minutes
Part B: $126.1575
Step-by-step explanation:
Part A,
From 7:45 to 12:15 is 4 hours and 30 minutes
From 12:45 which he clocked back in to 6:00 is 5 hours and 15 minutes.
4 hours and 30 minutes plus 5 hours and 15 minutes = 9 hours and 45 minutes.
Part B, 9 hours and 45 minutes can be represented as 9.45
And since he gets 15.35 "per" hour shows we can multiply.
15.35 x 9.45 = 126.1575
Therefor he earned 126.15.
Hey buddy its easy
Answer is C part
Pemdas
or
sadmep
it depends.
Parenthesis
exponents
multiplication
division
addition
Subtraction
![\bf \underset{dependent}{\stackrel{\textit{songs I can download}}{S(M)}}~~=~~\underset{independent}{\stackrel{\textit{how much memory I have}}{M}}](https://tex.z-dn.net/?f=%5Cbf%20%5Cunderset%7Bdependent%7D%7B%5Cstackrel%7B%5Ctextit%7Bsongs%20I%20can%20download%7D%7D%7BS%28M%29%7D%7D~~%3D~~%5Cunderset%7Bindependent%7D%7B%5Cstackrel%7B%5Ctextit%7Bhow%20much%20memory%20I%20have%7D%7D%7BM%7D%7D)
the memory is the independent, because that can vary at will, so it's independently varying at will, whilst the amount of songs are limited to how much memory there's in the CD, so the songs are dependent on the memory availability.