Solutions by induction

[Ducale]

Help with solutions by induction

Hi all,

Let me start of by saying that I haven't done any math like this since I was in first year (~6 years ago now) so I'm a little fuzzy... I mostly do stats.

I'm auditing a math bio course as part of my graduate thesis and I've done the first 2 questions on the assignment without any problems but this third question is stumping me because a) I have never done anything with induction and b) I haven't done integrals in 6 years!

Could someone help me through this step by step? The prof said that since this isn't really relevant to why I'm taking the course I could just differentiate and sub in the two differential equations, but it's been even longer since I've done anything like that!

Thanks!

 

[HallsofIvy]

Hi all,

Let me start of by saying that I haven't done any math like this since I was in first year (~6 years ago now) so I'm a little fuzzy... I mostly do stats.

I'm auditing a math bio course as part of my graduate thesis and I've done the first 2 questions on the assignment without any problems but this third question is stumping me because a) I have never done anything with induction and b) I haven't done integrals in 6 years!

Could someone help me through this step by step? The prof said that since this isn't really relevant to why I'm taking the course I could just differentiate and sub in the two differential equations, but it's been even longer since I've done anything like that!

Thanks!

View attachment 7612

If, as you appear to be saying, you do not know how to "differentiate and sub into the differential equations" you should not be taking this course.

For n= 0, you asked to show the function \(\displaystyle p_0(t)= \frac{(\mu t)^ne^{-\mu t}}{0!}= e^{-\mu t}\) satisfies the equation \(\displaystyle \frac{dp_0}{dt}= -\mu p_0(t)\) with the initial condition \(\displaystyle p_0(0)= 1\). Okay, with x= 0, \(\displaystyle p_0(0)= e^{-\mu (0)}= 1\) so the initial condition is satisfied. I would hope that you know that \(\displaystyle \frac{dp_0}{dt}= \frac{e^{-\mu t}}{dt}= -\mu e^{-\mu t}= -\mu p_0(t)\) so the statement is true for n= 0.

Now, to use the "induction" you want to prove that, if the statement is true for all n less than k then it is also true for k= n.

Given that \(\displaystyle p_k(t)= \frac{(\mu t)^k e^{\mu t}}{k!}\) then \(\displaystyle p_k(0)= \frac{(0)^ke^{0}}{k!}= 0\) so the initial condition is satisfied. Use the same formula to determine \(\displaystyle p_{k-1}(t)\) and \(\displaystyle p_{k-2}(t)\), calculate the derivative of \(\displaystyle p_k(t)\), put them all into the differential equation and show that it is satisfied.