Separation of Variables

Many differential equations are not solvable except by either numerical methods or by guess-and-check. Some, however, can be solved by separating the x's and y's, then integrating. This device cannot display Java animations. The above is a substitute static image
 See About the calculus applets for operating instructions.

The applet shows the slope field for dy/dx = y. The solution appears to be an exponential function.
In fact, if we guess that y = ex and plug that in, we find that it works I.E: We can be more methodical by using a technique called separation of variables. You start with the differential equation and use algebra to move the y's to the left-hand side and the x's to the right-hand side. You also treat dy and dx as if they were objects and move them, using algebra, to the appropriate sides. Then integrate both sides and simplify. For this example, it looks like In the first step, we divide both sides by y, then multiply both sides by dx. Integrating both sides by their respective variables yields the result.

Note that, since both C's are just arbitrary constants, we can subtract either one of them from both sides and then combine them into a new arbitrary constant to get Exponentiating both sides gives where we converted eC to a new arbitrary constant A>0. Getting rid of the absolute value just allows A to also be negative. Can A be 0? We need to check this case, because when we initially divided by y on both sides, we might have lost the solution y = 0. In fact, this is a solution, so we wind up with as the family of solutions for this differential equation, where A can be any real number.

Separation of variables only works if we can move the y's to the left-hand side using multiplication or division, not addition or subtraction. So something like dy/dx = x + y is not separable, but dy/dx = y + xy is separable, because we can factor the y out of the terms on the right-hand side, then divide both sides by y.

An equation like dy/dx = (x + 3)/(y - 2) is also separable, because we can multiply both sides by (y - 2); it is ok to move constants to either side. Your calculus textbook may have other examples of separable differential equations that you can type in to this applet and see what the graph looks like (remembering that if the solution isn't a function of x, the graphing algorithm may mess up). While you are here..

... I have a small favor to ask. Over the years we have used advertising to support the site so it can remain free for everyone. However, advertising revenue is falling and I have always hated the ads. So, would you go to Patreon and become a patron of the site? When we reach the goal I will remove all advertising from the site.

It only takes a minute and any amount would be greatly appreciated. Thank you for considering it!   – John Page

Become a patron of the site at   patreon.com/mathopenref

## Acknowledgements

Derived from the work of Thomas S. Downey under a Creative Commons Attribution 3.0 License.