Differential Equations Day 2 -- Separable Differential Equations
Separable Differential Equations
Today we'll learn about our first type of differential equation, a separable differential equation, and an algebraic method for solving it. When you're done with today you will know how to identify a separable first order differential equation, how to use the process to solve a separable differential equation, how to use GeoGebra to assist with the solution, and how to check that your solution is correct visually and algebraically.
Definition:
A separable differential equation is one that can be manipulated algebraically so that it is of the form:
Solution Process
To solve a separable differential equation, multiply both sides by dx to clear the dx in the denominator of dy/dx, and then integrate both sides.
After completing the integral (which might be quite difficult--use GeoGebra to assist!), it's typical to solve the resulting (non differential) equation for y to obtain the general solution of the differential equation.
The general solution of a differential equation is the solution that involves unspecified constants of integration. If a differential equation includes an initial condition then we can use the initial condition to obtain a specific solution. The difference between general and specific solutions will become clearer as we work through examples, so don't stress if you don't quite understand it yet.
A good reference on separable differential equations can be found here:
https://tutorial.math.lamar.edu/classes/de/separable.aspx
Examples
These are examples of separable differential equations.
Example 1 This differential equation from day 1 is separable.
However, it is not of the form at present. First you need to multiply both sides by y:
Now and , and so we can confirm that this is in fact a separable differential equation.
The next step is to multiply both sides by dx, and then we will integrate.
Notice two constants of integration. Since is equivalent to a single arbitrary constant of integration , we can subtract from both sides, and replace with .
Now, as stated above, solve for y, which means multiplying both sides by 2. Notice that is still an arbitrary constant, so we can replace with .
Lastly, take the square root of both sides to obtain the general solution
We can also just refer to as just c. In fact, whenever multiple constants of integration are absorbed into a single constant of integration, it's normal to simply refer to them as c. With this final tweak, we obtain the general solution. Noe that this is the same solution we saw on day 1; now we know how it was gotten!
The differential equation, in its original "slope field form" -- -- is shown in the GeoGebra applet below. The solution is also shown. Note that the + and the - parts of the square root of the solution are shown as two different functions and .
Example 2: The following is also a separable differential equation with "initial value".
,
We'll solve it together in class. See the notes for the worked solution.
The GeoGebra applet below shows the slope field of the differential equation. If you want, after we find the solution (or solutions), plot it (or them) in the applet.
Example 3:
We may not get to this in class. If so, then I will update this page with the solution. A slope field is visible below however.