| Differential Equations and Separation of Variables | More Practice |
| Slope Fields |
When you start learning how to integrate functions, you’ll probably be introduced to the notion of Differential Equations and Slope Fields.
Differential Equations and Separation of Variables
A differential equation is basically any equation that has a derivative in it. For example, a differential equation might include the variables $ x$, $ y$, and the derivative of $ y$ ($ {y}’$ or $ \displaystyle \frac{{dy}}{{dx}}$). Remember the following from the Antiderivatives and Indefinite Integration section:
To solve a differential equation, we need to separate the variables so that one variable is on the left-hand side of the equal sign (usually the “$ y$”, including the “$ dy$”), and the other variable is on the right-hand side (usually the “$ x$”, including the “$ dx$”). This is called separation of variables.
At this point, we can integrate on both sides, and put the “$ +\,C$” on the side with the $ x$’s (right away). We will simplify as needed, being careful to preserve the “$ +\,C$” during the simplification. Notice that if we add, subtract, multiply, or divide with the “$ +\,C$” when simplifying, we still end up with a constant $ (+\,C$). This is called the general solution.
Here are some examples:
Sometimes we have to solve the differential equation and then put in an initial value to get the $ “C”$ (particular solution). When doing this, it’s best to put in the initial condition immediately after differentiating. Here are some examples; sorry the math is so messy:
Here are a couple more likes ones we’ll see in the Exponential Growth Using Calculus section, where we have to set up and solve a differential equation that models a verbal statement:
Slope Fields
Slope Fields are a strange concept (they look funny!), but they really aren’t that difficult. They are little lines on a coordinate system graph that represent the slope for that $ (x,y)$ combination for a particular differential equation (remember that a differential equation represents a slope). For example, for the differential equation $ \displaystyle \frac{{dy}}{{dx}}=x+y$, for point $ (0,0)$ on the slope field graph, the little line would be horizontal, since $ 0+0=0$, and the slope of 0 is represented by a horizontal line.
Let’s show an example. For differential equation $ \displaystyle \frac{{dx}}{{dy}}=\frac{x}{y},$ the slope is 0 everywhere $ x$ is 0, and is 1 everywhere $ x$ and $ y$ are the same. Also, it is 2 when $ x$ is twice the value of $ y$, 3 when $ x$ is three times the value of $ y$, and so on. The slope is undefined (vertical lines or no lines) when $ y=0$. Thus, here’s a slope field. Again, remember that the little lines represent the slope, since a differential equation is a slope.
Here are more examples of slope fields. Note that if we solved the differential equation, we’d see the solution to that differential equation in the slope field pattern. For example, for the differential equation $ \displaystyle \frac{{dy}}{{dx}}=2$, the little lines in the slope field graph are $ \displaystyle y=2x$.
Here’s a few more that we’ll draw with solution curves, given the solution passes through the given initial points. To draw the solution curves, start with the initial point, and then follow the curve of the little lines as you best can:
Here’s a type of slope field problem you might see:
Learn these rules, and practice, practice, practice!
Use the Mathway keyboard to enter a Differential Equations problem, and then click on Submit (the arrow to the right of the problem) to solve the problem. You can also click on the 3 dots and then Examples in the upper right hand corner to drill down for example problems under “Differential Equations“.
If you click on “Tap to view steps”, you will go to the Mathway site, where you can register for the full version (steps included) of the software. You can even get math worksheets.
You can also go to the Mathway site here, where you can register, or just use the software for free without the detailed solutions. There is even a Mathway App for your mobile device. Enjoy!
On to L’Hopital’s Rule – you are ready!