What is the Sturm-Liouville Problem for $y''+2y'+ty=0$ with $y(0)=y(1)=0$?

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In summary: Then, with that definition, write down the inner product in question, and the corresponding eigenvalue problem.In summary, for the given differential equation $y''+2y'+ty=0$ on $0<x<1$ with boundary conditions $y(0)=y(1)=0$, the corresponding Sturm-Liouville operator is $L=-\frac{d}{dx}(e^{2x}\frac{d}{dx})$. Formulating the problem entails writing the equation in Sturm-Liouville form, which can be achieved by defining the weight function $r(x)=-e^{2x}$ and the eigenvalue $\lambda=-t$. The inner product is then defined as $\langle f,g \rangle
  • #1
Poirot1
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consider $y''+2y'+ty=0$ on $0<x<1$ such that $y(0)=y(1)=0$
Find the corresponding Sturm-Liouville operator and formulate the Sturm-Liouville problem. Hence, define the inner product for which the eigenfunctions are orthogonal.

I have $L=-e^-2x. d/dx(e^2x.d/dx)$ (how to do fractions in code?)

I'm not sure what is meant by formulate the problem but perhaps Ly=ty.

I don't know how to choose an inner product.

 
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  • #2
Question: is $t$ the variable of differentiation, or is it the eigenvalue?

You can do fractions with the \frac{numerator}{denominator} command.
 
  • #3
't' is the eigenvalue
 
  • #4
So, assuming you meant
$$L=-\frac{1}{e^{2x}}\,\frac{d}{dx}\left[e^{2x}\,\frac{d}{dx}\right],$$
I would agree that's the Sturm-Liouville operator.
In comparing it with the standard form of the Sturm-Liouville operator, what are $p, q,$ and $w$?
 
  • #5
$$p(x)=e^2x$$, $$q(x)=0$$,$$r(x)= -e^2x$$. What does formulating the problem entail? By your hint on the other thread, I might guess that I should define the inner product to be $$<f,g>= \int{ \frac{1}{r}*f*g$$.
 
  • #6
I don't know what your $r$ is - is that your weight function? Incidentally, you can write more than one thing in an exponent, in $\LaTeX$, if you enclose it in curly braces {} thus: $e^{2x}$. Compare with $e^2x$.
 
  • #7
yes, my r is your w.
 
  • #8
Poirot said:
yes, my r is your w.

So, in writing your original DE in Sturm-Liouville form, you have now discovered what the weight function is. If you look at my big post in your other thread, note how the weight function shows up in the inner product w.r.t. which the eigenfunctions of the Sturm-Liouville operator are orthogonal. It's not quite what you have in your post # 5 of this thread.
 
  • #9
So did formulating the problem simply mean writing in sturm-liouville form?
 
  • #10
Poirot said:
So did formulating the problem simply mean writing in sturm-liouville form?

Correct. Then, to solve the problem, you simply need to recognize what the weight function is, and write down the corresponding inner product.
 
  • #11
ok so <f,g> should be the integral of 'rgf' between 0 and 1 but $$r=-e^{-2x}$$ is negative. Should I take the mod?
 
  • #12
Yeah, you can't have a negative weight function, because then $\langle f|f\rangle$ might be negative, which is not allowed in an inner product. You should not take the mod, however. What I would do is absorb the minus sign into the eigenvalue. That is, define $\lambda=-t$ as the eigenvalue for the Sturm-Liouville operator.
 

FAQ: What is the Sturm-Liouville Problem for $y''+2y'+ty=0$ with $y(0)=y(1)=0$?

1. What is the Sturm-Liouville problem?

The Sturm-Liouville problem is a mathematical problem that involves finding the eigenvalues and eigenfunctions of a second-order linear differential equation. It is named after the mathematicians Jacques Charles François Sturm and Joseph Liouville, who first studied this problem in the 19th century.

2. What are the applications of the Sturm-Liouville problem?

The Sturm-Liouville problem has many applications in physics, engineering, and other areas of science. It is commonly used in solving problems related to heat transfer, fluid dynamics, quantum mechanics, and vibrating systems. It is also used in the study of partial differential equations and Fourier series.

3. How is the Sturm-Liouville problem solved?

The Sturm-Liouville problem is typically solved by finding the eigenvalues and eigenfunctions of the associated second-order linear differential equation. This can be done using various techniques such as separation of variables, Green's functions, and the spectral method. The solution to the problem is a set of eigenvalues and corresponding eigenfunctions, which form a complete orthogonal set.

4. What is the significance of the eigenvalues and eigenfunctions in the Sturm-Liouville problem?

The eigenvalues and eigenfunctions in the Sturm-Liouville problem have important physical and mathematical significance. The eigenvalues represent the possible frequencies at which a system can vibrate, while the eigenfunctions correspond to the shape of the vibration at a particular frequency. They also play a crucial role in solving boundary value problems and understanding the behavior of linear differential equations.

5. Are there any limitations to the Sturm-Liouville problem?

While the Sturm-Liouville problem is a powerful tool for solving many mathematical and physical problems, it does have some limitations. It is only applicable to linear second-order differential equations with specific boundary conditions. It also assumes that the coefficients in the differential equation are continuous. In some cases, the problem may not have a solution, or the solution may not be unique.

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