Answer:
y(t) = c₁ e^(-1/2 t) cos(√3/2 t) + c₂ e^(-1/2 t) sin(√3/2 t) + 1
Step-by-step explanation:
y" + y' + y = 1
This is a second order nonhomogenous differential equation with constant coefficients.
First, find the roots of the complementary solution.
y" + y' + y = 0
r² + r + 1 = 0
r = [ -1 ± √(1² − 4(1)(1)) ] / 2(1)
r = [ -1 ± √(1 − 4) ] / 2
r = -1/2 ± i√3/2
These roots are complex, so the complementary solution is:
y = c₁ e^(-1/2 t) cos(√3/2 t) + c₂ e^(-1/2 t) sin(√3/2 t)
Next, assume the particular solution has the form of the right hand side of the differential equation. In this case, a constant.
y = c
Plug this into the differential equation and use undetermined coefficients to solve:
y" + y' + y = 1
0 + 0 + c = 1
c = 1
So the total solution is:
y(t) = c₁ e^(-1/2 t) cos(√3/2 t) + c₂ e^(-1/2 t) sin(√3/2 t) + 1
To solve for c₁ and c₂, you need to be given initial conditions.
Answer:
5/11 is your answer.
Step-by-step explanation:
What you do is you need to find a number that goes in to both 125 and 275.
5 goes in to both of them.
125/5=25
275/5=55
25/55 This can still be reduced by 5.
25/5=5
55/5=11
5/11 is your answer.
Using the combination formula, it is found that six of these five-ball selections contain exactly five red balls.
The order in which the balls are selected is not important(as balls A, B, C, D and E is the same outcome as balls B, A, C, D and E), hence the <em>combination formula</em> is used to solve this question. If the order was important, then the permutation formula would be used.
<h3>What is the combination formula?</h3>
is the number of different combinations of x objects from a set of n elements, given by:
In this problem, five red balls can be chosen from a set of six, hence the number of selections is the combination of 5 elements from a set of 6, that is calculated from the formula given above:
Hence six of these five-ball selections contain exactly five red balls.
More can be learned about the combination formula at brainly.com/question/25821700
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