A .
this is false
There’s no answer for x and y
Answer:
slope is (y1-y2)/x1-x2 pick two points on the line
Step-by-step explanation:
R is the rate (10%)
C=20, begiing rate
T=time=8
so
h=20(1+0.10)^8
h=20(1.1)^8
h=42.8718
ronnd
$42.87
D is answer
Note that if
![{x_2}'=x_1](https://tex.z-dn.net/?f=%7Bx_2%7D%27%3Dx_1)
, then
![{x_2}''={x_1}'](https://tex.z-dn.net/?f=%7Bx_2%7D%27%27%3D%7Bx_1%7D%27)
, and so we can collapse the system of ODEs into a linear ODE:
![{x_2}''=3{x_2}'-x_2+e^t](https://tex.z-dn.net/?f=%7Bx_2%7D%27%27%3D3%7Bx_2%7D%27-x_2%2Be%5Et)
![{x_2}''-3{x_2}'+x_2=e^t](https://tex.z-dn.net/?f=%7Bx_2%7D%27%27-3%7Bx_2%7D%27%2Bx_2%3De%5Et)
which is a pretty standard linear ODE with constant coefficients. We have characteristic equation
![r^2-3r+1=\left(r-\dfrac{3+\sqrt5}2\right)\left(r+\dfrac{3+\sqrt5}2\right)=0](https://tex.z-dn.net/?f=r%5E2-3r%2B1%3D%5Cleft%28r-%5Cdfrac%7B3%2B%5Csqrt5%7D2%5Cright%29%5Cleft%28r%2B%5Cdfrac%7B3%2B%5Csqrt5%7D2%5Cright%29%3D0)
so that the characteristic solution is
![{x_2}_C=C_1e^{(3+\sqrt5)/2\,t}+C_2e^{-(3+\sqrt5)/2\,t}](https://tex.z-dn.net/?f=%7Bx_2%7D_C%3DC_1e%5E%7B%283%2B%5Csqrt5%29%2F2%5C%2Ct%7D%2BC_2e%5E%7B-%283%2B%5Csqrt5%29%2F2%5C%2Ct%7D)
Now let's suppose the particular solution is
![{x_2}_p=ae^t](https://tex.z-dn.net/?f=%7Bx_2%7D_p%3Dae%5Et)
. Then
![{x_2}_p={{x_2}_p}'={{x_2}_p}''=ae^t](https://tex.z-dn.net/?f=%7Bx_2%7D_p%3D%7B%7Bx_2%7D_p%7D%27%3D%7B%7Bx_2%7D_p%7D%27%27%3Dae%5Et)
and so
![ae^t-3ae^t+ae^t=-ae^t=e^t\implies a=-1](https://tex.z-dn.net/?f=ae%5Et-3ae%5Et%2Bae%5Et%3D-ae%5Et%3De%5Et%5Cimplies%20a%3D-1)
Thus the general solution for
![x_2](https://tex.z-dn.net/?f=x_2)
is
![x_2=C_1e^{(3+\sqrt5)/2\,t}+C_2e^{-(3+\sqrt5)/2\,t}-e^t](https://tex.z-dn.net/?f=x_2%3DC_1e%5E%7B%283%2B%5Csqrt5%29%2F2%5C%2Ct%7D%2BC_2e%5E%7B-%283%2B%5Csqrt5%29%2F2%5C%2Ct%7D-e%5Et)
and you can find the solution
![x_1](https://tex.z-dn.net/?f=x_1)
by simply differentiating
![x_2](https://tex.z-dn.net/?f=x_2)
.