Having problems solving the Linear Homogeneous Differential Equation. Will post Equation on nest post using Equations plug-in.

Question

anonymous · Answer

$$\frac{ dN }{ dt }=nf \prime(N^{*}) $$

anonymous · Answer

This should be (by definition) exp(f'(N*)t) However my work shows me differently

dumbcow · Answer

what does n refer to? a variable or constant

anonymous · Answer

That is where I am steering wrong. N is a function of t. The solution shows n(t )= exp(f'(N*)t) I tried to solve this with both n being a variable and constant. I still dont have their solution.

dumbcow · Answer

hmm ok well assuming "n" is same as "N" in that its the  dependent variable
$$\frac{dN}{dt} = N*f'(N*)$$
$$\frac{dN}{N} = f'(N*) dt$$
integrate both sides
$$\ln (N) = f'(N*) t + C$$
$$N = e^{f'(N*)t +C} = k e^{f'(N*)}$$

anonymous · Answer

Well... that does work. And it's the solution that I expect. My problem was not identifying the dependent variable. I originally integrated f'(N*) as a function of dN. That's why I was getting screwy answers. I appreciate your insight and help.

dumbcow · Answer

your welcome

ahh i see...yeah always look at what you are integrating with respect to..."dt" or "dN"

anonymous · Answer

I am usually fine with these mechanically but when working on independent research I sometimes don't see the correct solution to the DE. I really appreciate your help. I will pay moer attention to what is the variable in question.