Answer:
k = 30, ![y(t) = C_1e^{5t}+C_2e^{6t}](https://tex.z-dn.net/?f=y%28t%29%20%3D%20C_1e%5E%7B5t%7D%2BC_2e%5E%7B6t%7D)
Step-by-step explanation:
Since
is a solution, then it must satisfy the differential equation. So, we calculate the derivatives and replace the value in the equation. We have that
![\frac{d^2y}{dt^2} = 25 e^{5t},\frac{dy}{dt} = 5e^{5t}](https://tex.z-dn.net/?f=%5Cfrac%7Bd%5E2y%7D%7Bdt%5E2%7D%20%3D%2025%20e%5E%7B5t%7D%2C%5Cfrac%7Bdy%7D%7Bdt%7D%20%3D%205e%5E%7B5t%7D)
Then, replacing the derivatives in the equation we have:
![25e^{5t}-11(5)e^{5t}+ke^{5t}=0 e^{5t}(25-55+k) =0](https://tex.z-dn.net/?f=25e%5E%7B5t%7D-11%285%29e%5E%7B5t%7D%2Bke%5E%7B5t%7D%3D0%20e%5E%7B5t%7D%2825-55%2Bk%29%20%3D0)
Since
is a positive function, we have that
.
Now, consider a general solution
, then, by calculating the derivatives and replacing them in the equation, we get
![Ae^{rt}(r^2-11r+30)=0](https://tex.z-dn.net/?f=Ae%5E%7Brt%7D%28r%5E2-11r%2B30%29%3D0)
We already know that r=5 is a solution of the equation, then we can divide the polynomial by the factor (r-5) to the get the other solution. If we do so, we get that (r-6)=0. So the other solution is r=6.
Therefore, the general solution is
![y(t) = C_1e^{5t}+C_2e^{6t}](https://tex.z-dn.net/?f=y%28t%29%20%3D%20C_1e%5E%7B5t%7D%2BC_2e%5E%7B6t%7D)
The answer to this question is FALSE.
Domain is the set of all the numbers that we can input to the function or that can be used in place of x. The numbers which make the function undefined are excluded from the domain.
In the given exponential function, there is no any value of x which will make the function undefined, so the domain of the function if set of All real numbers. In general, domain of exponential functions is Set of All real numbers.
Yes because if you turn 6 and three to 12 then you have 2/12 and 4/12!
Answer:
Step-by-step explanation:
![y=-x^{2} -5x-6](https://tex.z-dn.net/?f=y%3D-x%5E%7B2%7D%20-5x-6)
![y=-(x^2+5x+6)](https://tex.z-dn.net/?f=y%3D-%28x%5E2%2B5x%2B6%29)
![y=-(x+3)(x+2)](https://tex.z-dn.net/?f=y%3D-%28x%2B3%29%28x%2B2%29)
Set each factors equal to zero.
![(x+3)=0 , x=-3](https://tex.z-dn.net/?f=%28x%2B3%29%3D0%20%2C%20%20x%3D-3)
![x+2=0 , x=-2](https://tex.z-dn.net/?f=x%2B2%3D0%20%20%2C%20%20%20x%3D-2)
The x-int is
![(-2,0)](https://tex.z-dn.net/?f=%28-2%2C0%29)