Determine whether y = xn is a solution to the d.e.
x2y" + ny = n2y.
When y = xn, we have y' = nxn - 1 and
y" = n(n - 1)x(n - 2).
Then the left-hand side of the d.e. is
x2y " + ny = x2[n(n - 1)x(n - 2)] + n[xn]
= n(n - 1)xn + nxn
= n2xn - nxn + nxn
The right-hand side of the d.e. is
n2y = n2xn.
Since the left- and right- hand sides of the d.e. are the same, y = xnis a solution to this differential equation.
You may also encounter problems that look something like this:
Show that the function such-and-such is a solution to the d.e. blah-blah."
While the word "show" may strike terror in your heart, don't give in to fear! Problems that say "show" are actually easier than ones that look like this:
"Determine whether the function such-and-such is a solution to the d.e. blah-blah."
Regardless of whether a problem says "show" or "determine", do the same thing: evaluate the left-hand side of the d.e., evaluate the right-hand side of the d.e., and see if you get the same thing from both sides.
Here's the difference between these types of problems: If a problem says "determine," you need to figure out whether the function is a solution to the d.e. or not. If a problem says "show," it's telling you the answer you'll get if you do the work right.