Calculus Blog

Well, thinking that we have a test on slope fields as well as a few things with differential equations...I figured I should go through the steps of doing that.

Basically, if they ask you to consider a differential equation (also known as the equation that will give you a slope at a certain x and y), then they would ask to sketch the slope field on axes at given points. All you have to do is take each individual point and plug in the x and the y into the differential equation to get the slope at each point. Then you need to draw little lines...basically a /, |,_, or \ at each given point to indicate either positive slope, undefined slope, a slope of 0, or a negative slope respectively. Also, if you have a slope of 2 and a slope of 1, the slope of 2 should show up a bit more steep than the slope of 1.

Another type of question given with these is to find the particular solution to the differential equation given a specific condition, such that f(x) = c. Basically all you need to do is to work the equation around so that all the x's and dx are on one side and all of the y's and dy's are on the other side. Once you have that, integrate both sides of the equation. When it asks you about a particular solution, it's asking basically about the value of "C" whenever you integrate it so that it fits the conditions. So, we integrate both sides accordingly and put the +c on the side with the x. Now we need to solve the equation for y. After solving the equation for y, you must now solve for c after you plug in for the conditions it gave you (the x and the y value). After solving for c, you need to plug this back into our equation that was solved for y. Now simplify the equation and that is your answer :-).

Now that you have completed the particular solution...you may be asked to do something such as the the limit of it or asked maybe even a concavity or increasing/decreasing question (which you can do, now that you have the equation).

Hope this helps someone study if you look at it in time :o.

-John