Examples of differential equations, with rules for their solution

At this point, please just believe this. You will be able to verify this for yourself in a couple of sections.

Solving a differential equation

Now all we need to do is apply the initial conditions. This means that we need the derivative of the solution.

For a rare few differential equations we can do this. However, for the vast majority of the second order differential equations out there we will be unable to do this. So, we would like a method for arriving at the two solutions we will need in order to form a general solution that will work for any linear, constant coefficient, second order homogeneous differential equation. This is easier than it might initially look. We will use the solutions we found in the first example as a guide.

All of the solutions in this example were in the form. The important idea here is to get the exponential function. Once we have that we can add on constants to our hearts content. To see if we are correct all we need to do is plug this into the differential equation and see what happens. This can be reduced further by noting that exponentials are never zero. Okay, so how do we use this to find solutions to a linear, constant coefficient, second order homogeneous differential equation?

Once we have these two roots we have two solutions to the differential equation. Well here we would just use the reverse power rule. We would increment the exponent, so it's Y to the first, but so now when we take the anti-derivative, it will be Y squared, and then we divide by that incremented exponent, is equal to, well the exciting thing about E to the X is that it's anti-derivative, and its derivative, is E to the X, is equal to E to the X, plus is equal to E to the X plus C.

And so we can leave it like this if we like. In fact this right over here is, this isn't an explicit function. Y here isn't an explicit function of X.

We could actually say Y is equal to the plus or minus square root of two times all of this business, but this would be a pretty general relationship, which would satisfy this separable differential equation. Let's do another example. So let's say that we have the derivative of Y with respect to X is equal to, let's say it's equal to Y squared times sine of X. Pause the video and see if you can find the general solution here.

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2. Differential Equations - Separable Equations;
3. 1. Solving Differential Equations.
4. Shattered.
5. Finding general solutions using separation of variables!

So once again, we wanna separate our Ys and our Xs. So let's see, we can multiply both sides times Y to the negative two power, Y to the negative two, Y to the negative two, these become one, and then we could also multiply both sides times DX. So if we multiply DX here, those cancel out, and then we multiply DX here, and so we're left with Y to the negative two power times DY is equal to sine of X, DX, and now we just can integrate both sides.

Now what is the anti-derivative of Y to the negative two? The following examples show how to solve differential equations in a few simple cases when an exact solution exists. A separable linear ordinary differential equation of the first order must be homogeneous and has the general form.

We may solve this by separation of variables moving the y terms to one side and the t terms to the other side ,. We solve the transformed equation with the variables already separated by Integrating ,. Then, by exponentiation , we obtain. It is easy to confirm that this is a solution by plugging it into the original differential equation:.

One must also assume something about the domains of the functions involved before the equation is fully defined. The solution above assumes the real case. First-order linear non-homogeneous ODEs ordinary differential equations are not separable. They can be solved by the following approach, known as an integrating factor method.

Consider first-order linear ODEs of the general form:.