Linear Differential Operator/Inner Product

In summary, the inner product defined on the vector space V of all real polynomials by <f,g>=Int(x*e^(-x)*f(x)*g(x),x=o..infinity is shown to be symmetric with respect to the linear differential operator L=-x*(d^2/dx^2)-(2-x)*(d/dx):V->V. This is done by replacing f(x) in the integral with Lf(x) and showing that <Lf, g>= <f, Lg>.
  • #1
Akers
2
0
Can anyone help me?

Q. An inner product is defined on the vector space V of all (real) polynomials (of arbitrary
degree) by

<f,g>=Int(x*e^(-x)*f(x)*g(x),x=o..infinity

Show that the linear differential operator

L=-x*(d^2/dx^2)-(2-x)*(d/dx):V->V

is symmetric with respect to this inner product.Don't necessarily need the exact answer, just need to know how I'm meant to go about it.
 
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  • #2
Akers said:
Can anyone help me?

Q. An inner product is defined on the vector space V of all (real) polynomials (of arbitrary
degree) by

<f,g>=Int(x*e^(-x)*f(x)*g(x),x=o..infinity

Show that the linear differential operator

L=-x*(d^2/dx^2)-(2-x)*(d/dx):V->V

is symmetric with respect to this inner product.


Don't necessarily need the exact answer, just need to know how I'm meant to go about it.

Let f and g be a polynomials. Show that <Lf, g>= <f, Lg>.
 
  • #3
HallsofIvy said:
Let f and g be a polynomials. Show that <Lf, g>= <f, Lg>.

Ah right, thanks.

So all I need to do is replace the f(x) in the integral with Lf(x) (and the same with g(x) on the other side)? I'll give it a go anyway. Cheers.
 

1. What is a linear differential operator?

A linear differential operator is a mathematical function that operates on a function by taking its derivatives. It can be represented as a polynomial of derivatives, with coefficients that may be functions of the independent variable.

2. How is a linear differential operator different from a regular function?

Unlike a regular function, a linear differential operator operates on a function by taking its derivatives. This means that the output of a linear differential operator is also a function, rather than a single value.

3. What is an inner product in the context of linear differential operators?

An inner product is a mathematical operation that takes two functions as inputs and produces a single value as output. In the context of linear differential operators, the inner product is used to measure the similarity between two functions.

4. How is the inner product used in solving differential equations?

The inner product is used in solving differential equations by providing a way to determine the coefficients of the linear differential operator. By finding the inner product of the operator and the functions involved in the equation, the coefficients can be solved for.

5. Can linear differential operators be used in other areas of science?

Yes, linear differential operators have applications in many areas of science, including physics, engineering, and economics. They are commonly used to model and solve complex systems and processes that involve rates of change.

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