![\dfrac{\mathrm dy}{\mathrm dt}=-4x\implies x=-\dfrac14\dfrac{\mathrm dy}{\mathrm dt}\implies\dfrac{\mathrm dx}{\mathrm dt}=-\dfrac14\dfrac{\mathrm d^2y}{\mathrm dt^2}](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20dy%7D%7B%5Cmathrm%20dt%7D%3D-4x%5Cimplies%20x%3D-%5Cdfrac14%5Cdfrac%7B%5Cmathrm%20dy%7D%7B%5Cmathrm%20dt%7D%5Cimplies%5Cdfrac%7B%5Cmathrm%20dx%7D%7B%5Cmathrm%20dt%7D%3D-%5Cdfrac14%5Cdfrac%7B%5Cmathrm%20d%5E2y%7D%7B%5Cmathrm%20dt%5E2%7D)
Substituting this into the other ODE gives
![-\dfrac14\dfrac{\mathrm d^2y}{\mathrm dt^2}=-4y\implies y''-16y=0](https://tex.z-dn.net/?f=-%5Cdfrac14%5Cdfrac%7B%5Cmathrm%20d%5E2y%7D%7B%5Cmathrm%20dt%5E2%7D%3D-4y%5Cimplies%20y%27%27-16y%3D0)
Since
, it follows that
. The ODE in
has characteristic equation
![r^2-16=0](https://tex.z-dn.net/?f=r%5E2-16%3D0)
with roots
, admitting the characteristic solution
![y_c=C_1e^{4t}+C_2e^{-4t}](https://tex.z-dn.net/?f=y_c%3DC_1e%5E%7B4t%7D%2BC_2e%5E%7B-4t%7D)
From the initial conditions we get
![y(0)=5\implies 5=C_1+C_2](https://tex.z-dn.net/?f=y%280%29%3D5%5Cimplies%205%3DC_1%2BC_2)
![y'(0)=16\implies-16=4C_1-4C_2](https://tex.z-dn.net/?f=y%27%280%29%3D16%5Cimplies-16%3D4C_1-4C_2)
![\implies C_1=\dfrac12,C_2=\dfrac92](https://tex.z-dn.net/?f=%5Cimplies%20C_1%3D%5Cdfrac12%2CC_2%3D%5Cdfrac92)
So we have
![\boxed{y(t)=\dfrac12e^{4t}+\dfrac92e^{-4t}}](https://tex.z-dn.net/?f=%5Cboxed%7By%28t%29%3D%5Cdfrac12e%5E%7B4t%7D%2B%5Cdfrac92e%5E%7B-4t%7D%7D)
Take the derivative and multiply it by -1/4 to get the solution for
:
![-\dfrac14y'(t)=\boxed{x(t)=-\dfrac12e^{4t}+\dfrac92e^{-4t}}](https://tex.z-dn.net/?f=-%5Cdfrac14y%27%28t%29%3D%5Cboxed%7Bx%28t%29%3D-%5Cdfrac12e%5E%7B4t%7D%2B%5Cdfrac92e%5E%7B-4t%7D%7D)
Multiply <span>3/8 inch by 75 and divide by 5</span>
Answer:
216x
Step-by-step explanation:
- Write it out:
- Add in x: 216x
I hope this helps!
If s represents the number of songs Kim downloads, her expenses for a month must satisfy the inequality
5.00 + 0.58·s ≤ 13.00
To solve this, subtract 5.00 to get the s-term by itself, then divide by the coefficient of s.
0.58·s ≤ 8.00
s ≤ 8.00/0.58 ≈ 13.79
The number of songs must be an integer, so we conclude Kim can download 13 songs, but not 14 songs.
Kim can download 13 songs without exceeding her budget.
$793.50, All you do is multiply $15.87 by 50