2nd order DEQ: weird solution method

fusi0n

1. The problem statement, all variables and given/known data

Suppose that u(t) is a solution to

y'' + p(t)y' + q(t)y = 0

Suppose a second solution has the form

y(t) = m(t)u(t)

where m(t) is an unknown function of t. Derive a first order linear differential equation for m'(t).

Suppose y(t) = e^(2t). Use the method above to find a second solution to this equation.

3. The attempt at a solution

I tried:

y = m*u
=> m = y/u
=> m' = (y'/u) - (y/(u^2))*u'

There isn't anything in my book or in my class notes that describes the technique to be used here. Does anyone have any suggestions on how to get started?

Mindscrape

Have you heard of variation of parameters? Take a look, and see if it helps at all.

http://en.wikipedia.org/wiki/Variation_of_parameters

gammamcc

Surely this is recommended where e^{2t} is a solution to the original 2nd order diff. eq. Right? Then, like Mindscrape says, it's variation of parameters.

Mathgician

try using this equation:

Y2=y1*integral{[e^(-integral p(t))dt]}/[(y1)^2]dt

y1=y(t)=e^(2t)

then the general solution is: y=c1y1+c2y2

Mathgician

And also to add, your method works best when the p(t)y' value is excluded from the equation. You can solve your problem by the method you mentioned above, but it cost much time and leaves much room for mistakes, and you are essentially guessing by trial and error by multiplying u(t) by respective powers of x, or m, as you would put it. So therefore, for the sake of your precious time and of doing unneeded amount of work, you should try this equation. Tell me how it works, I believe you will learn to like it. And, later you will learn to solve these problems without the guesswork, but the problems get very tedious, believe me.... I'm not a big fan of solving these equations by hand.

fusi0n

this is an inhomogeneous differential equation. the method of variation of parameters is for homogeneous. in the wikipedia article it is said to "determine the homogeneous solution using a method of your choice"

as for as the other method is concerned: how can I use this method to derive a general first order linear differential equaiton for m'(t)? by the way mathgician, this method works great for finding the second solution!!!

fusi0n

never mind, mathgician, i out the way to find m'(t). your input helped a lot. i will post my solution to the problem if anyone is interested.