I'll assume the ODE is
Solve the homogeneous ODE,
The characteristic equation
has roots at and
. Then the characteristic solution is
For nonhomogeneous ODE (1),
consider the ansatz particular solution
Substituting this into (1) gives
For the nonhomogeneous ODE (2),
take the ansatz
Substitute (2) into the ODE to get
Lastly, for the nonhomogeneous ODE (3)
take the ansatz
and solve for .
Then the general solution to the ODE is
Answer:
a)
Operating room x 1 2 3 4
P(x) 0.15 0.25 0.4 0.2
b)
The graph of probability distribution is in attached image.
c)
All the probabilities lies in the range (0 to 1) and the probabilities add up to 1 so, the computed probability distribution is the valid probability distribution.
Step-by-step explanation:
Number of operating rooms Days
1 3
2 5
3 8
4 4
a)
Sum of frequencies=1+2+3+4=10
Operating rooms x frequency f Relative frequency
1 3 3/20=0.15
2 5 5/20=0.25
3 8 8/20=0.4
4 4 4/20=0.2
So, using the relative frequency approach, a discrete probability distribution for number of operating rooms in use on any given day is
Operating room x 1 2 3 4
P(x) 0.15 0.25 0.4 0.2
b)
The graph of probability distribution is in attached image.
c)
There are two conditions for a probability distribution to be valid
1. All probability must ranges from 0 to 1.
2. All probabilities must add up to 1.
We can see that all the probabilities lies in the range (0 to 1), so, condition 1 is satisfied.
For condition 2,
sum[p(x)]= 0.15+0.25+0.4+0.2=0.4+0.6=1.
As the probabilities add up to 1, so the condition 2 is also satisfied.
Thus, the computed probability distribution is the valid probability distribution.
