Question

Second Order Differential Equation with Non-constant Coefficients (question will be typed below)

Answer 1

\[\frac{ d^2y }{ dx^2 }-\frac{ dy }{ dx }-(\frac{ 6 }{ x^2 }+\frac{ 2 }{ x })y = 0\]

Answer 2

I think I'm supposed to use reduction of order but I'm not exactly sure where to start

Answer 3

Here are some notes/examples for reduction of order
http://tutorial.math.lamar.edu/Classes/DE/ReductionofOrder.aspx

Answer 4

I believe you will need to start with an initial solution of the form:
\[y = ax^2 + c\]

Answer 5

The problem says I'm looking for a value of lambda such that

y = x^lambda

is a solutino

Answer 6

If I'm not given an initial solution how do I find one?

Answer 7

yeah thats the tricky part, usually by an educated guess then checking to see if it works.
since the non-constant coefficient has an x^2 in denominator, it makes sense for y to have an x^2 term. Also this allows for 1st derivative to be linear which will cancel out the "2y/x" part of coeffecient.

Answer 8

thank you, that's the part I was having some trouble with

I'll give it a shot and see how it goes

Answer 9

good luck, do you want me to post the solution or you can check answer on wolfram too ?

Answer 10

I tried wolfram but it ended up giving me a pretty complex solution

Answer 11

looks like i was wrong then.
Initial solution should be of the form
y = c/x^2 or y= x^-2

Then try to get 2nd solution by solving for function v(x)
y = v*x^-2

Answer 12

I got the answer, thank you for your help!

I did a substitution of y = x^lambda in the original equation