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Differential Equations - Concept
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
Differential equations are equations with a derivative of an unknown function. Solving a differential equation requires using antidifferentiation. Since they use antiderivatives, there are multiple solutions. Differential equations can be classified by their order, which is the same as the largest derivative in the equation (1st, 2nd, etc.).
I want to talk about a new concept, the concept of differential equation. Let's look at a definition. An equation containing the derivatives of an unknown function's called an differential equation. And I have some examples here. dy dx equals 2x. Here the unknown function would be y. The second derivative of y with respect to x is 6x+8. Here the second derivative minus x times the first derivative plus the function itself equals 0. And so on.
Differential equations are classified according to order, and the order of a differential equation is determined by the highest order derivative in the equation. So this would be a first order because you have a first order derivative here. This is a second order differential equation because you have a second order. This has a first and a second order, so it's going to be a second order differential equation, also a second order. And another first order. And of course you can have there's no limit the order of a differential equation but we'll focus on first and second order ones and pretty simple ones too.
Let's talk about what a solution is. A differential equation, a solution the differential equation is a function that satisfies the equation. And what we're going to be trying to do when we're solving differential equations is finding al the solutions. So let's look at an example. This is the first in our list of examples dy dx equals 2x. So this is a function whose derivative at every point has a equals to x. So we want to find out what kind of function that is. Well, whenever you have information about functions derivative you can get information about the function by anti-differentiating. If we're told the derivative is 2x then the function itself y is the integral of 2x dx. Now, if the derivative was with respect to x we would integrate with respect to x. And of course we can solve this.
So y=x squared plus c, right? The anti-derivative of 2x is x squared plus c. So this describes all of the functions that satisfy this differential equation. They're all parabolas. Now, if you were to draw a picture let's just go over here really quickly. Parabolas looking like this, like this, like this. These would all be solutions to this original differential equation. We call something like this the general solution of the differential equation because in one formula we have every single solution. We get a different solution for every value of c that we plug in here. Now, for example c could be 0. y=x squared to solution this differential equation. But y=x squared plus 100 is also a solution. And so in one formula we have for representation of all the solutions, all infinite of them. So that's called a general solution, the differential equation.and that's going to be one of our goals when we're solving differential equations.
So what we just saw is that a differential equation has many solutions and many solutions can sometimes be written in a single for called a general solution. We had this differential equation dy dx equals 2x and that has a general solution y=x squared plus c.
There's another type of problem that you may see in your homework where you're given a differential equation and an initial condition y=0 when x=-3 and this can help narrow it down. Among all these general solution you can narrow it down to a particular solution. One particular solution that passes through this point. And so from the differential equation we would get the general solution y=x squared plus c and then we'd use this initial condition, y=0 when x=-3. So plug in -3, y would be 0 and this helps us solve for the value of c. The particular value of c that we need for this particular solution. This is going to be 9+c=0. So c=-9. And so our particular solution would be y=x squared minus 9. It's one of these in the general solutions. One particular example of the general solution. So this is a particular solution to our differential equation with initial condition.
So let's just sum up. 2 kinds of problems that you can see in your homework. First a differential equation, find the general solution. This represents infinitely many different solutions. You could also have a differential equation plus initial condition and it will in general have one particular solution and so you might have a differential equation and this is what an initial condition looks like. It's a condition on y. Sometimes it's a condition on the derivative, y=0 when x=-3. This is called a particular solution to this initial value problem, differential equation plus initial condition.
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