12 3/8. Feel free to ask any questions!
Answer:
The general solution of the given differential equation
![y = ( c_{1} + c_{2} x ) cos3 x + ( c_{3} +c_{4} x) sin3 x](https://tex.z-dn.net/?f=y%20%3D%20%20%28%20c_%7B1%7D%20%2B%20c_%7B2%7D%20x%20%29%20cos3%20x%20%2B%20%28%20%20c_%7B3%7D%20%2Bc_%7B4%7D%20x%29%20sin3%20x)
Step-by-step explanation:
Step(I):-
Given differential equation
y⁴+18y"+81y=0
⇒ (D⁴+18D²+81)y =0
The auxiliary equation
![m^4+18m^2+81 =0](https://tex.z-dn.net/?f=m%5E4%2B18m%5E2%2B81%20%3D0)
![(m^2)^{2} + 2 (9) m^{2} +(9)^2 = 0](https://tex.z-dn.net/?f=%28m%5E2%29%5E%7B2%7D%20%2B%202%20%289%29%20m%5E%7B2%7D%20%2B%289%29%5E2%20%3D%200)
we will use formula ( a + b)² = a² + 2 a b + b²
⇒ ( m² + 9 ) ² = 0
⇒ ( m² + 9 ) ( m² + 9 ) = 0
![m^{2} =-9\\m= - 3i and m=3i](https://tex.z-dn.net/?f=m%5E%7B2%7D%20%3D-9%5C%5Cm%3D%20-%203i%20and%20m%3D3i)
m² + 9 = 0
![m² = -9\\m= -3i and m=3i](https://tex.z-dn.net/?f=m%C2%B2%20%3D%20-9%5C%5Cm%3D%20-3i%20and%20m%3D3i)
The complex roots are 0± 3 i ,0 ± 3 i
<em>Step(ii)</em>:-
The complementary function
![y = e^{\alpha x } ( c_{1} + c_{2} x ) cos\beta x + ( c_{3} +c_{4} x) sin\beta x](https://tex.z-dn.net/?f=y%20%3D%20e%5E%7B%5Calpha%20x%20%7D%20%28%20c_%7B1%7D%20%2B%20c_%7B2%7D%20x%20%29%20cos%5Cbeta%20x%20%2B%20%28%20%20c_%7B3%7D%20%2Bc_%7B4%7D%20x%29%20sin%5Cbeta%20x)
The general solution of the given differential equation
![y = e^{0 x } ( c_{1} + c_{2} x ) cos3 x + ( c_{3} +c_{4} x) sin3 x](https://tex.z-dn.net/?f=y%20%3D%20e%5E%7B0%20x%20%7D%20%28%20c_%7B1%7D%20%2B%20c_%7B2%7D%20x%20%29%20cos3%20x%20%2B%20%28%20%20c_%7B3%7D%20%2Bc_%7B4%7D%20x%29%20sin3%20x)
The general solution of the given differential equation
18 ft because area is length times width which is 2x9 in this situation and it equal 18 giving you the answer.
Answer:
3
Step-by-step explanation:
the absolute meaning of a number is never a negative number
Answer: 16,384
Step-by-step explanation:
The seven marbles have different colours, so we can differentiate them.
Now, suppose that for each marble we have a selection, where the selection is in which jar we put it.
For the first marble, we have 4 options ( we have 4 jars)
For the second marble, we have 4 options.
Same for the third, for the fourth, etc.
Now, the total number of combinations is equal to the product of the number of options for each selection.
We have 7 selections and 4 options for each selection, then the total number of combinations is:
C = 4^7 = 16,384