I was able reach patial solution and hope you would help to prove these two questions, thank you! (1 point) A Bernoulli differential equation is one of the formdy+ P(x)y = Q(x)y".dxObserve that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, thesubstitution u = yl-n transforms the Bernoulli equation into the linear equationdu+ (1-п)P(u)и — (1 - п)Q(m).daUse an appropriate substitution to solve the equation- -U =and find the solution that satisfies y(1) = 1.y(z) = (1 point) The radioactive isotope carbon-14 is present in small quantities in all life forms, andit is constantly replenished until the organism dies, after which it decays to stable carbon-12at a rate proportional to the amount of carbon-14 present, with a half-life of 5550 years.Suppose C(t) is the amount of carbon-14 present at time t.(a) Find the value of the constant k in the differential equation C" = -kC.k =(b) In 1988 three teams of scientists found that the Shroud of Turin, which was reputed to bethe burial cloth of Jesus, contained about 91 percent of the amount of carbon-14 containedin freshly made cloth of the same material[1]. How was old the Shroud of Turin in 1988,according to these data?Age =years[1]: The New York Times, October 18, 1988.

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Description: I was able reach patial solution and hope you would help to prove these two questions, thank you! (1 point) A Bernoulli differential equation is one of the formdy+ P(x)y = Q(x)y".dxObserve that, if n = 0 or 1, the Bernoulli equation is linear. For ot

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(1 point) A Bernoulli differential equation is one of the form dy + P(x)y = Q(x)y". dx Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u = yl-n transforms the Bernoulli equation into the linear equation du + (1-п)P(u)и — (1 - п)Q(m). da Use an appropriate substitution to solve the equation - -U = and find the solution that satisfies y(1) = 1. y(z) =

(1 point) A Bernoulli differential equation is one of the form dy + P(x)y = Q(x)y". dx Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u = yl-n transforms the Bernoulli equation into the linear equation du + (1-п)P(u)и — (1 - п)Q(m). da Use an appropriate substitution to solve the equation - -U = and find the solution that satisfies y(1) = 1. y(z) =

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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I was able reach patial solution and hope you would help to prove these two questions, thank you!

(1 point) A Bernoulli differential equation is one of the form
dy
+ P(x)y = Q(x)y".
dx
Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the
substitution u = yl-n transforms the Bernoulli equation into the linear equation
du
+ (1-п)P(u)и — (1 - п)Q(m).
da
Use an appropriate substitution to solve the equation
- -U =
and find the solution that satisfies y(1) = 1.
y(z) =

(1 point) The radioactive isotope carbon-14 is present in small quantities in all life forms, and
it is constantly replenished until the organism dies, after which it decays to stable carbon-12
at a rate proportional to the amount of carbon-14 present, with a half-life of 5550 years.
Suppose C(t) is the amount of carbon-14 present at time t.
(a) Find the value of the constant k in the differential equation C" = -kC.
k =
(b) In 1988 three teams of scientists found that the Shroud of Turin, which was reputed to be
the burial cloth of Jesus, contained about 91 percent of the amount of carbon-14 contained
in freshly made cloth of the same material[1]. How was old the Shroud of Turin in 1988,
according to these data?
Age =
years
[1]: The New York Times, October 18, 1988.

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