# Reduction od order question

1. Aug 17, 2010

### beetle2

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

find the first and second derivative of first solution.

2. Relevant equations

$y(x)=m(x)y_1(x)$

$y'(x)=m'(x)y_1(x)+m(x)y_1'(x)$

3. The attempt at a solution

I have been given $y_1=\frac{1}{x^n}$

Which part is the m(x) and which is $y_1(x)$
I'm not sure how to do the substitution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Aug 17, 2010

### hunt_mat

Can you give us a little more of the question.

3. Aug 17, 2010

### beetle2

Sure the original question is Differential equation.

It asks find a pair of fundamentle soultions to the DE

$x^2y''-3xy'+y=0$

So I'm trying to find the two solutions I'm taking $y_1=\frac{1}{x^n}$
as one solution.

4. Aug 17, 2010

### hunt_mat

This is known as Eulers equation, you are correct (partially) in looking for solutions of the form:
$$y=x^{n}$$
Insert this into your ODE and you will obtain a quadratic equation for $$n$$, this will have two solutions which correspond to the two solutions.

5. Aug 17, 2010

### beetle2

So does

$x^2y''-3xy'+y=0$

become

$x^2(n^2-2)x^{n-2} -3xnx^{n-1} + x^n=0$

6. Aug 17, 2010

### beetle2

I seem to be having trouble getting a quadratic for n

is this the right method?

I have

$y=x^n$
than
$y'=nx^{n-1}$
then
$y''=(n^2-n-1)x^{n-2}$

so do you substitute into

$x^2y''-3xy'+y=0$ to give

$x^2((n^2-n-1)x^{n-2})- 3x(nx^{n-1})+x^n=0$

I'm getting

$(n^2-4n+1)x^n$

I can see a characteristic equation there but not sure about the coefficient $x^n$?

7. Aug 18, 2010

### hunt_mat

Almost:
$$y''=n(n-1)x^{n-2}$$
So
$$x^{2}(n(n-1)x^{n-2}-3xnx^{n-1}+x^{n}=0$$
which yields
$$(n^{2}-4n+1)x^{n}=0$$
So
$$n^{2}-4n+1=0$$
Can you calculate n from the above?

8. Aug 18, 2010

### beetle2

Thanks,

I solve the quadratic $$n^{2}-4n+1=0$$
and get:

$$n_1=-\sqrt{3}-2$$ and $$n_2=\sqrt{3}+2$$

So are the two solutions....?

$$y_1=c_1x^{-\sqrt{3}-2}$$ and $$y_2=c_2x^{\sqrt{3}+2}$$

Last edited: Aug 18, 2010
9. Aug 18, 2010

### vela

Staff Emeritus
$$y_1=c_1x^{-\sqrt{3}+2}$$