In mathematics, function<span> composition is the pointwise application of one </span>function<span> to the result of another to produce a third </span>function<span>. ... Intuitively, </span>composing two functions<span> is a chaining process in which the output of the inner </span>function<span> becomes the input of the outer </span>function<span>.</span>
One year equals 31536000 seconds
365×24×60×60
Answer:
![y(x)=e^{-2x}[3cos(\sqrt{6}x)+\frac{2\sqrt{6}}{3}sin(\sqrt{6}x)]](https://tex.z-dn.net/?f=y%28x%29%3De%5E%7B-2x%7D%5B3cos%28%5Csqrt%7B6%7Dx%29%2B%5Cfrac%7B2%5Csqrt%7B6%7D%7D%7B3%7Dsin%28%5Csqrt%7B6%7Dx%29%5D)
(See attached graph)
Step-by-step explanation:
To solve a second-order homogeneous differential equation, we need to substitute each term with the auxiliary equation 
where the values of 
are the roots:
Since the values of are complex conjugate roots, then the general solution is ![y(x)=e^{\alpha x}[C_1cos(\beta x)+C_2sin(\beta x)]](https://tex.z-dn.net/?f=y%28x%29%3De%5E%7B%5Calpha%20x%7D%5BC_1cos%28%5Cbeta%20x%29%2BC_2sin%28%5Cbeta%20x%29%5D)
where 
.
Thus, the general solution for our given differential equation is ![y(x)=e^{-2x}[C_1cos(\sqrt{6}x)+C_2sin(\sqrt{6}x)]](https://tex.z-dn.net/?f=y%28x%29%3De%5E%7B-2x%7D%5BC_1cos%28%5Csqrt%7B6%7Dx%29%2BC_2sin%28%5Csqrt%7B6%7Dx%29%5D)
.
To account for both initial conditions, take the derivative of 
, thus, ![y'(x)=-2e^{-2x}[C_1cos(\sqrt{6}x+C_2sin(\sqrt{6}x)]+e^{-2x}[-C_1\sqrt{6}sin(\sqrt{6}x)+C_2\sqrt{6}cos(\sqrt{6}x)]](https://tex.z-dn.net/?f=y%27%28x%29%3D-2e%5E%7B-2x%7D%5BC_1cos%28%5Csqrt%7B6%7Dx%2BC_2sin%28%5Csqrt%7B6%7Dx%29%5D%2Be%5E%7B-2x%7D%5B-C_1%5Csqrt%7B6%7Dsin%28%5Csqrt%7B6%7Dx%29%2BC_2%5Csqrt%7B6%7Dcos%28%5Csqrt%7B6%7Dx%29%5D)
Now, we can create our system of equations given our initial conditions:
![y(x)=e^{-2x}[C_1cos(\sqrt{6}x)+C_2sin(\sqrt{6}x)]\\\\y(0)=e^{-2(0)}[C_1cos(\sqrt{6}(0))+C_2sin(\sqrt{6}(0))]=3\\\\C_1=3](https://tex.z-dn.net/?f=y%28x%29%3De%5E%7B-2x%7D%5BC_1cos%28%5Csqrt%7B6%7Dx%29%2BC_2sin%28%5Csqrt%7B6%7Dx%29%5D%5C%5C%5C%5Cy%280%29%3De%5E%7B-2%280%29%7D%5BC_1cos%28%5Csqrt%7B6%7D%280%29%29%2BC_2sin%28%5Csqrt%7B6%7D%280%29%29%5D%3D3%5C%5C%5C%5CC_1%3D3)
![y'(x)=-2e^{-2x}[C_1cos(\sqrt{6}x+C_2sin(\sqrt{6}x)]+e^{-2x}[-C_1\sqrt{6}sin(\sqrt{6}x)+C_2\sqrt{6}cos(\sqrt{6}x)]\\\\y'(0)=-2e^{-2(0)}[C_1cos(\sqrt{6}(0))+C_2sin(\sqrt{6}(0))]+e^{-2(0)}[-C_1\sqrt{6}sin(\sqrt{6}(0))+C_2\sqrt{6}cos(\sqrt{6}(0))]=-2\\\\-2C_1+\sqrt{6}C_2=-2](https://tex.z-dn.net/?f=y%27%28x%29%3D-2e%5E%7B-2x%7D%5BC_1cos%28%5Csqrt%7B6%7Dx%2BC_2sin%28%5Csqrt%7B6%7Dx%29%5D%2Be%5E%7B-2x%7D%5B-C_1%5Csqrt%7B6%7Dsin%28%5Csqrt%7B6%7Dx%29%2BC_2%5Csqrt%7B6%7Dcos%28%5Csqrt%7B6%7Dx%29%5D%5C%5C%5C%5Cy%27%280%29%3D-2e%5E%7B-2%280%29%7D%5BC_1cos%28%5Csqrt%7B6%7D%280%29%29%2BC_2sin%28%5Csqrt%7B6%7D%280%29%29%5D%2Be%5E%7B-2%280%29%7D%5B-C_1%5Csqrt%7B6%7Dsin%28%5Csqrt%7B6%7D%280%29%29%2BC_2%5Csqrt%7B6%7Dcos%28%5Csqrt%7B6%7D%280%29%29%5D%3D-2%5C%5C%5C%5C-2C_1%2B%5Csqrt%7B6%7DC_2%3D-2)
We then solve the system of equations, which becomes easy since we already know that 
:
Thus, our final solution is:
![y(x)=e^{-2x}[C_1cos(\sqrt{6}x)+C_2sin(\sqrt{6}x)]\\\\y(x)=e^{-2x}[3cos(\sqrt{6}x)+\frac{2\sqrt{6}}{3}sin(\sqrt{6}x)]](https://tex.z-dn.net/?f=y%28x%29%3De%5E%7B-2x%7D%5BC_1cos%28%5Csqrt%7B6%7Dx%29%2BC_2sin%28%5Csqrt%7B6%7Dx%29%5D%5C%5C%5C%5Cy%28x%29%3De%5E%7B-2x%7D%5B3cos%28%5Csqrt%7B6%7Dx%29%2B%5Cfrac%7B2%5Csqrt%7B6%7D%7D%7B3%7Dsin%28%5Csqrt%7B6%7Dx%29%5D)
Answer:
12
Step-by-step explanation:
In four acres there are 3 gardens so you just have to multiply 4x3=12
Answer:
C. y = 2x + 1.50
Step-by-step explanation:
Initial fee is $1.50, this does not change.
For every mile it cost $2 times however many mile(s) is traveled.
So, initial fee + (mile cost * miles travel)
