Let be a positive number. A differential equation of the form dy/dy = ky^1 + c w

Question

Let be a positive number. A differential equation of the form dy/dy = ky^1 + c where k is a positive constant, is called a doomsday equation because the exponent in the expression ky^1+ c is larger than the exponent 1 for natural growth. (a) Determine the solution that satisfied the initial condition y(0) = y_0. (b) Show that there is a finite time t = T (doomsday) such that lim_t = T y(t) = infinity. (c) An especially prolific breed of rabbits has the growth term ky^1.01. If 2 such rabbits breed initially and the warren has 16 rabbits after three months, then when is doomsday?

Explanation / Answer

Solution:

(a)

dy/dt = k*y^(1 + c)

separate and integrate

y^-(1+c) dy = k dt ...

integrate using the standard rule for x^n

(y^(-c))/(-c) = kt + C

... multiply both sides by -c, note that any changes to C don't matter yet since C can be anything

y^-c = -ckt + D...

take both sides to the power of -1/c

y = (-ckt + D)^(-1/c)

now we need to solve for D, given that y(0) = y0

y0 = (0 + D)^(-1/c)... take both sides to the -c

D = (y0)^-c

So your answer is:

y = (-ckt + y0^(-c))^(-1/c)

Part b:

note that c is positive, which means the exponent -1/c is negative.

Taking zero to an infinite power results in an infinite discontinuity i.e. your "doomsday".

So just find the value of t when the base is equal to zero:

-ckt + y0^(-c) = 0

y0^(-c) = ckt

t = y0^(-c) / ck

That t is your T, which is to say your Doomsday

c) in this case c = 0.01. Your Y0=2.

Use the fact that there are 16 rabbits after three months (that is, y(3 months) = 16 )

in order to solve for k.

y = (-ckt + y0^(-c))^(-1/c)

16 =(-0.01 *3*k + 2^(-0.01))(-1/0.01)

k = 0.68

your Doomsday

t = y0^(-c) /ck

t = 2(-0.01) / (-0.01 * 0.68)

t = 146 months

Answer

Related Questions

Navigate

• Browse (All)
• Browse L
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.