Consider the following differential equation: dxdt=k(4−x)(1−x)

Question

dx/dt = k(4-x)(1-x)

Is this ordinary DE or partial DE?

State the order?

What are the independent variable(s)?

What are the dependent variable(s)?

If the equation is ordinary, is it linear or nonlinear?

I believe it is ordinary and first order, I am not sure if it is linear or not. I think the dependent variable is x but I am confused.

Answer 1

If the solution sought is x(t) then you are right, t is independent and x dependent.

dx/dt = k(4-x)(1-x)

same as x'-k(4-x)(1-x)-0,,,,,first order derivative (isolated so linear), but dependent variable is squared so non-linear. If there are just two variables the equation is ordinary differential equation.

by way, isn't this seperable and solution is (1/3)ln(4-x)-(1/3)ln(1-x)=kt+C

and in a way I could say t(x)=(1/k)*[(1/3)ln((4-x)/(1-x))-C is solution and t is dependent and x independent.

And if the equation is written dt/dx-(1/k)*1/[(4-x)*(1-x)]=0, then it would be first order linear since neither t' or t is in non-linear form.