Determine whether *y* = *x*^{n} is a solution to the d.e.

*x*^{2}*y*" + *ny* = *n*^{2}*y*.

Answer

When *y* = *x*^{n}, we have *y*' = *nx*^{n - 1} and

*y*" = *n*(*n* - 1)*x*^{(n - 2)}.

Then the left-hand side of the d.e. is

*x*^{2}y " + *ny* = *x*^{2}[*n*(*n* - 1)*x*^{(n - 2)}] + *n*[*x*^{n}]

= *n*(*n* - 1)*x*^{n} + *nx*^{n}

= *n*^{2}*x*^{n} - *nx*^{n} + *nx*^{n}

= *n*^{2}*x*^{n}.

The right-hand side of the d.e. is

*n*^{2}y = *n*^{2}*x*^{n}.

Since the left- and right- hand sides of the d.e. are the same, *y* = *x*^{n} *is* 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.