How to Solve and Apply Boundary Conditions for the Sturm-Liouville Problem

[ftarak]

Homework Statement

the question is how to write the following equation to S-L form and solve the equation

Homework Equations

$$x^{2}y'' + xy' + \lambda^{2}y = 0$$

$$y(1/10) = y(2) = 0$$

The Attempt at a Solution

I tried to write above equation in form

$$d[x^{2}y']/dx + 2\lambda^{2}y = 0 <br />$$
but it doesn't work, because it is not equal the above equation. Does anybody know how to re-write above equation in S-L form?

 

[snipez90]

Divide the original equation through by x and note that xy'' + y' = (xy')'.

 

[vela]

If you have a differential equation of the form

p₀(x) y'' + p₁(x) y' + p₂(x) y = 0

multiply it by

$$\frac{1}{p_0(x)}\exp\left[\int^x \frac{p_1(x')}{p_0(x')}\,dx'\right]$$

Then you'll be able to write the first two terms in the form you want.

 

[ftarak]

Divide the original equation through by x and note that xy'' + y' = (xy')'.

thank you so much. it was so helpful

 

[ftarak]

If you have a differential equation of the form

p₀(x) y'' + p₁(x) y' + p₂(x) y = 0

multiply it by

$$\frac{1}{p_0(x)}\exp\left[\int^x \frac{p_1(x')}{p_0(x')}\,dx'\right]$$

Then you'll be able to write the first two terms in the form you want.

thanks buddy

 

[ftarak]

hi guys,
I have another question, when I suppose that one of the solution for this problem is

$$y(x) = x^{m}$$
actually, it doesn't work because it directs me to trivial solution and gives me the zero coefficients after applying the boundary conditions, do you have any idea about the supposition of yj(x) for applying the boundary condition and then calculating the coefficients.

many thanks if you give me a hint