Answer:
The two solutions are given as
and ![y_2(t)=\dfrac{-1}{9}e^{-8t}+\dfrac{1}{9}e^{t}](https://tex.z-dn.net/?f=y_2%28t%29%3D%5Cdfrac%7B-1%7D%7B9%7De%5E%7B-8t%7D%2B%5Cdfrac%7B1%7D%7B9%7De%5E%7Bt%7D)
Step-by-step explanation:
As the given equation is
![y''+7y'-8y=0\\](https://tex.z-dn.net/?f=y%27%27%2B7y%27-8y%3D0%5C%5C)
So the corresponding equation is given as
![m^2+7m-8=0](https://tex.z-dn.net/?f=m%5E2%2B7m-8%3D0)
Solving this equation yields the value of m as
![(m+8)(m-1)=0\\m=-8, m=1](https://tex.z-dn.net/?f=%28m%2B8%29%28m-1%29%3D0%5C%5Cm%3D-8%2C%20m%3D1)
Now the equation is given as
![y(t)=C_1e^{m_1t}+C_2e^{m_2t}](https://tex.z-dn.net/?f=y%28t%29%3DC_1e%5E%7Bm_1t%7D%2BC_2e%5E%7Bm_2t%7D)
Here m1=-8, m2=1 so
![y(t)=C_1e^{-8t}+C_2e^{t}](https://tex.z-dn.net/?f=y%28t%29%3DC_1e%5E%7B-8t%7D%2BC_2e%5E%7Bt%7D)
The derivative is given as
![y'(t)=-8C_1e^{-8t}+C_2e^{t}](https://tex.z-dn.net/?f=y%27%28t%29%3D-8C_1e%5E%7B-8t%7D%2BC_2e%5E%7Bt%7D)
Now for the first case y(t=0)=1, y'(t=0)=0
![y(t=0)=C_1e^{-8*0}+C_2e^{0}\\1=C_1+C_2\\\\y'(t=0)=-8C_1e^{-8*0}+C_2e^{0}\\0=-8C_1+C_2](https://tex.z-dn.net/?f=y%28t%3D0%29%3DC_1e%5E%7B-8%2A0%7D%2BC_2e%5E%7B0%7D%5C%5C1%3DC_1%2BC_2%5C%5C%5C%5Cy%27%28t%3D0%29%3D-8C_1e%5E%7B-8%2A0%7D%2BC_2e%5E%7B0%7D%5C%5C0%3D-8C_1%2BC_2)
So the two equation of co-efficient are given as
![C_1+C_2=1\\-8C_1+C_2=0](https://tex.z-dn.net/?f=C_1%2BC_2%3D1%5C%5C-8C_1%2BC_2%3D0)
Solving the equation yield
![C_1=1/9 \\C_2=8/9](https://tex.z-dn.net/?f=C_1%3D1%2F9%20%5C%5CC_2%3D8%2F9)
So the function is given as
![y_1(t)=\dfrac{1}{9}e^{-8t}+\dfrac{8}{9}e^{t}](https://tex.z-dn.net/?f=y_1%28t%29%3D%5Cdfrac%7B1%7D%7B9%7De%5E%7B-8t%7D%2B%5Cdfrac%7B8%7D%7B9%7De%5E%7Bt%7D)
Now for the second case y(t=0)=0, y'(t=0)=1
![y(t=0)=C_1e^{-8*0}+C_2e^{0}\\0=C_1+C_2\\\\y'(t=0)=-8C_1e^{-8*0}+C_2e^{0}\\1=-8C_1+C_2](https://tex.z-dn.net/?f=y%28t%3D0%29%3DC_1e%5E%7B-8%2A0%7D%2BC_2e%5E%7B0%7D%5C%5C0%3DC_1%2BC_2%5C%5C%5C%5Cy%27%28t%3D0%29%3D-8C_1e%5E%7B-8%2A0%7D%2BC_2e%5E%7B0%7D%5C%5C1%3D-8C_1%2BC_2)
So the two equation of co-efficient are given as
![C_1+C_2=0\\-8C_1+C_2=1](https://tex.z-dn.net/?f=C_1%2BC_2%3D0%5C%5C-8C_1%2BC_2%3D1)
Solving the equation yield
![C_1=-1/9 \\C_2=1/9](https://tex.z-dn.net/?f=C_1%3D-1%2F9%20%5C%5CC_2%3D1%2F9)
So the function is given as
![y_2(t)=\dfrac{-1}{9}e^{-8t}+\dfrac{1}{9}e^{t}](https://tex.z-dn.net/?f=y_2%28t%29%3D%5Cdfrac%7B-1%7D%7B9%7De%5E%7B-8t%7D%2B%5Cdfrac%7B1%7D%7B9%7De%5E%7Bt%7D)
So the two solutions are given as
and ![y_2(t)=\dfrac{-1}{9}e^{-8t}+\dfrac{1}{9}e^{t}](https://tex.z-dn.net/?f=y_2%28t%29%3D%5Cdfrac%7B-1%7D%7B9%7De%5E%7B-8t%7D%2B%5Cdfrac%7B1%7D%7B9%7De%5E%7Bt%7D)
A) 90-55=35 since a right angle is 90 total\
b)ii
Answer:
B) 3/9
Step-by-step explanation:
Whats the question if you dont give us a question we cant answer it
The answer is B. 20 is what number minus 6
“20 is what minus 6”
is = another word for equal to
minus = - or subtraction
20 = d - 6