In this video I go over a general overview on differential equations as well as a few of the definitions and terms associated with them. I discuss briefly about the order of a differential equation as well as what the solution to a differential equation is. I also allude to how most real-world models of differential equations are complex and thus don't usually have explicit solutions to them. Instead we often have to approximate the solutions and I will show how in later videos so stay tuned!
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Modeling with Differential Equations: General Differential Equations
In general, a differential equation is an equation that contains an unknown function and one or more of its derivatives.
The order of a differential equation is the order of the highest derivative that occurs in the equation.
Thus the differential equations in my earlier videos on population growth are first-order equations because they contain only the first derivative.
The differential equation developed for the motion of a spring, also in my previous videos, is a second-order equation because it contained a second derivative.
In those previous equations, the independent variable was t and represents time, but in general the independent variable doesn't have to represent time.
For example we consider the differential equation:
y' = xy
This equation is understood as y being the unknown function of x.
A function is called a solution of a differential equation if the equation is satisfied when y =f(x) and its derivatives are substituted into the differential equation.
Thus, f is a solution of the above differential equation if:
f'(x) = xf(x)
for all values of x in some interval.
When we are asked to solve a differential equation we are expected to find all possible solutions of the equation.
We have already solved some particularly simply differential equations, namely, those of the form:
y' = f(x).
For instance, we know that the general solution of the differential equation:
y' = x3
is given by:
Note that in this example, it appears that the differential y' = x3 doesn't contain the function y but it indeed can be written as:
And to clarify further, the order is the highest derivative that appears with a non-zero coefficient since technically you can write:
But, in general, solving a differential equation is not an easy matter.
There is no systematic technique that enables us to solve all differential equations.
In my later videos, I will show how we can draw rough graphs of solutions even when we have no explicit formula.
Also in later videos, I will go over how to find numerical approximations to solutions.