mjbc said:Homework Statement
dx/dt = k(a-x)(b-x)
x(0)=0
The Attempt at a Solution
dx/((a-x)(b-x)) = kdt
Integral (dx/(ab-bx-ax+x^2)) = kt +C
x/ab - (1/b)ln|x| - (1/a)ln|x| - (x^-1) = kt + C
however, if i try to substitute x(0) = 0, i get the ln 0
please help i have no idea if i am doing this right
The variable x represents the quantity being measured, such as the amount of a substance in a chemical reaction. t represents time. k, a, and b are constants that determine the rate of change and equilibrium points of the system.
This is a first-order, non-linear differential equation, meaning that the highest power of x is 1 and it is not directly proportional to t. It is also an autonomous equation, as the right-hand side does not explicitly depend on t.
The solution to this equation represents the behavior of a system over time, such as the concentration of a chemical in a reaction or the population of a species. It can also help predict the stability and equilibrium points of the system.
Differential equations are used in various fields of science, engineering, and economics to model the behavior of complex systems. This specific equation can be used to study chemical reactions, population dynamics, and other systems with changing quantities over time.
Some common methods for solving this type of differential equation include separation of variables, substitution, and Euler's method. More advanced techniques such as Laplace transforms and numerical methods may also be used.