
where can be linear, nonlinear, have constant coefficients or variable coefficients. Often times is a function of . That is, . The highest order derivative in the equation determines the order of the differential equation. Differential equations with only a single independent variable are called Ordinary Differential Equations (ODEs). Those with more than one independent variable are called Partial Differential Equations (PDEs), due to the fact that the derivatives are partial derivatives. The solution to a differential equation is the unknown function that you have the derivative for. While this may seems backward, in nature, we can observe how something changes over time, and thus, we can fit a derivative to this data. From this derivative, we then try to determine the original function. A differential equation describes changes in one variable relative to another variable, and as such, solutions to differential equations are functions that describe the ups and downs of a function. For example, or where is the dependent variable and is the independent variable. The differential equation: is an equation that says the change in for a certain change in is negatively proportional to the value of . In other words, when is large, the slope of the solution (d/d) is negative and proportional to (the proportionality constant is ). As becomes smaller, the slope becomes smaller. The solution to Equation 2.4.1 is where the constant is determined by the ``initial condition'', the value of when . If , then ; if , then . The solution of the differential equation, produces a ``class'' of similar solutions, and a particular member of that class is identified by the initial condition. Building an ODEno_title
Example 2.4.0.2 (Building an ODE)
Consider a simple chemical reaction. We have a substance, , that
spontaneously converts to with a rate, , while
spontaneously converts to with a rate, .
Schematically, we can notate this with the equation:
From this, we can describe the change in as the proportion of that converts into minus the proportion of that changes into . That is, If we enforce conservation of mass so that the combined mass of and is always constant, , we can now rewrite the Equation 2.4.3 as Without explicitly finding a solution to Equation 2.4.4, we can determine what it will be when it is at equilibrium. That is to say, we can determine what proportion of needs to be comprised of such that the amount of converting to is the same as the amount of converting to , or . We do this by setting the slope of to zero and solving for : Thus, if then the amounts of and will not change. Now that we know what the equilibrium is, it is interesting to look at the general solution to Equation 2.4.4 because the equilibrium plays a large role in it. Equation 2.4.4 can be solved using various methods. In Example 2.10.3.1 we show how to use an integrating factor to solve for and the result is: Notice what happens as gets larger and larger. If we take the limit, we get and thus, the system converges on the equilibrium. The exponential term simply causes the difference between the initial condition, the amount of at time , or , the equilibrium to become smaller and smaller as time passes. Two Componants as Oneno_title
Example 2.4.0.4 (Two Componants as One)
Now, consider a two component reaction,
This is called a second order reaction because the reaction rate depends on the concentration of two components, and . However, under certain conditions, it can be treated as a first order reaction, like in Example 2.4.0.1. When the concentration of or is essentially infinite, and there is a small concentration of the other component, then we have a pseudo first order reaction. Here we will show how this is possible from the differential equation. We start with a 2nd order equation where the rate of formation is determined by the concentration of [] and [], We assume that s collide with s at a rate determined by the temperature of the reaction and that a certain fraction of the collisions will result in making a . If the availability of is infinite so that its concentration never changes, the rate constant can be rewritten as a pseudo rate constant : and this allows us to treat the second order reaction as if it were first order. This assumption, that is infinite, is often reasonable when is some sort of drug compound and is a cellular receptor for this compound.
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