Home Articles Calculus Electromagnetism Relativity Physical Units Space-Time Part I:   Curved Space-Time Space-Time Part II:   Black Holes References Links Art Quotes Recipes Meat Marinades Al Pastor Carnitas Frijoles: Refried Beans Shrimp Burritos Shrimp Enchiladas Spanish Rice Cioppino Calzone or Pizza Korma Chicken Kung Pao Shrimp Thai Chicken and Bamboo Killer Shrimp Seafood Chowder Salmon Grilled Shrimp Spicy Tuna Herb Salad Wisconsin Bratwurst Cole Slaw Soup with Dumplings Garlic Cheese Sauce Hot Garlic Butter Greek Salad Leg of Lamb Cornbread Chocolate Fruit & Nut Pie Chocolate Truffles Rice Pudding Quick Chocolate Sauce Quick Chocolate Pie Cups-tsp-tbl-ml Ant Killer I recommend FireFox

A Basic Introduction to Differential Equations

Like integrals, differential equations are solved mostly by guessing. Unfortunately, there is little in the way of hard and fast rules. When we're solving a differential equation, we're looking for some function, F, that fits some particular prescription. The simpliest differential equation is:

d F
--- = 0
d x

This differential equation has as its only solution F(x) = constant. You can see this solution works by plugging the function into the equation and checking.

This differential equation is called a 1st order equation - there's only 1st order derivatives. Like polynomials, we're looking for as many solutions as there are orders. When we find that many solutions, we're done: the complete solution to the differential equation is the weighted sum of the seperate solutions.

The next simpliest differential equation is

d F
--- = k
d x

This equation has as its solution F = kx + c. Again, you can plug this in and try it out.

Next,

d F
--- = F
d x

That is, the derivative of the function is the function. We recognize the answer to this, F = e^x. There's no rule that makes this obvious; we just remember, in exactly the same way we remember things when we do a crossword puzzle.

A very important pair of differential equations are:
 

d2F ---- = k2F dx2

d2F ---- = -k2F dx2

These second-order equations come up in physics all the time. It's important to recognize them instantly and see the various solutions instantly. The solution to the first equation is F = ae^kx + be^-kx. These solutions have the rather unfortunate property that they're neither manifestly symmetric nor anti-symmetric. This can be easily fixed, however. Any function F(x) can be resolved into a symmetric and an anti-symmetric part,

F_symmetric = ( F(x) + F(-x) ) / 2

F_{anti-symmetric} =  ( F(x) - F(-x) ) / 2

F = F_symmetric + F_{anti-symmetric}

We can play the same trick with the solutions to the differential equation above,

F_symmetric = (e^kx + e^-kx) / 2 = cosh( kx )

F_{anti-symmetric} =  = (e^kx - e^-kx) / 2 = sinh( kx )

Now our solutions will be F = a cosh( kx ) + b sinh( kx ).

The solution to the second equation is  F = ae^ikx + be^-ikx. Again, these solutions are neither manifestly symmetric nor anti-symmetric. We can fix this with the same trick,

F_symmetric = (e^ikx + e^-ikx) / 2 = cos( kx )

F_{anti-symmetric} =  = (e^ikx - e^-ikx) / 2i = sin( kx )

Now our solutions will be F = a cos( kx ) + b sin( kx ).