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Annapetrik
20.03.2020 •
Mathematics
Suppose that r1 and r2 are roots of ar2 + br + c = 0 and that r1 = r2; then exp(r1t) and exp(r2t) are solutions of the differential equation ay + by + cy = 0. Show that φ (t; r1, r2) = er2t − er1t r2 −
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Ответ:
The Correct Question is:
Suppose that r1 and r2 are roots of ar² + br + c = 0 and that r1 = r2; then e^(r1t) and e^(r2t) are solutions of the differential equation
ay'' + by' + cy = 0.
Show that
φ (t; r1, r2) = [e^(r2t) - e^(r1t )]/(r2 - r1)
is a solution of the differential equation.
φ (t; r1, r2) is a solution of the differential equation, and it shown.
Step-by-step explanation:
Given the differential equation
ay'' + by' + cy = 0
and r1 and r2 are the roots of its auxiliary equation.
We want to show that
φ (t; r1, r2) = [e^(r2t) - e^(r1t )]/(r2 - r1)
satisfies the given differential equation, that is
aφ'' + bφ' + cφ = 0 (*)
Where φ = φ (t; r1, r2)
We now differentiate φ twice in succession, with respect to t.
φ' = [r2e^(r2t) - r1e^(r1t )]/(r2 - r1)
φ'' = [r2²e^(r2t) - r1²e^(r1t )]/(r2 - r1)
Using these in (*)
We have
a[r2e^(r2t) - r1e^(r1t )]/(r2 - r1) + [r2²e^(r2t) - r1²e^(r1t )]/(r2 - r1) + c[e^(r2t) - e^(r1t )]/(r2 - r1)
= [(ar2² + br2 + c)e^(r2t) - (ar1² + br1 + c)e^(r1t)]/(r1 - r2)
We know that r1 and r2 are the roots of the auxiliary equation
ar² + br + c = 0
and r1 = r2
This implies that
ar1² + br1 + c = ar2² + br2 + c = 0
And hence,
[(ar2² + br2 + c)e^(r2t) - (ar1² + br1 + c)e^(r1t)]/(r1 - r2) = 0
Therefore,
aφ'' + bφ' + cφ = 0
Ответ:
The firm can interpret the soil test by using Bayes’ Theorem to see what the posterior probabilities of seeing different of oil as wlel as no oil are. By using this, you can tell that it’s more likely that they are going to find medium quality oil.
P(E1soil)= .5*.2=.1
P(E2 soil)=.2*.8=.16
P(E3soil)=.3*.2=.06
P(soil)=.1+.16+.06=.32
P(E1|soil)=.1/.32=.3125
P(E2|soil)=.16/.32=.5
P(E3|soil)=.06/.32=.1875
P(Oil)=P(medium quality oil high quality oil)=.3125+.5=.81257
Hope this helps, now you know the answer and how to do it. HAVE A BLESSED AND WONDERFUL DAY! As well as a great rest of Black History Month! :-)
- Cutiepatutie ☺❀❤