Second order homog. DE non-const coeff.

In summary, the conversation is about a person who needs help verifying if a given function is in the kernel of a given differential operator. They have a second order homogeneous non-constant coefficients linear differential equation, but they can't remember how to solve it or if they even did in the past. They looked through the book, but it only covers a specific case. The person has made some progress so far, but they are stuck with a differential equation. The conversation ends with a recommendation to get some sleep.
  • #1
EvLer
458
0
I have a 2nd order homogeneous non-const. coefficients linear DE, and don't remember how we used to solve it or even if we did at all, looked through the book, but it only covers a case of Cauchy-Euler.

The original question actually goes like this:
verify that y(x) = sin (x2) is in the kernel of L,
L = D2 - x-1D + 4x2, where D is a differetiation operator.

so what I have so far is this:
Ly = 0
when I distribute I get this DE and get stuck with it:

y'' - x-1y' + 4x2y = 0

Thanks for any help.
 
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  • #2
EvLer said:
The original question actually goes like this:
verify that y(x) = sin (x2) is in the kernel of L,
L = D2 - x-1D + 4x2, where D is a differetiation operator.

so what I have so far is this:
Ly = 0
when I distribute I get this DE and get stuck with it:

y'' - x-1y' + 4x2y = 0

Thanks for any help.
This is a very simple question, just insert sin(x^2) for y.

ehild
 
  • #3
shoot...i need sleep. :yuck:

Thanks :smile:
 
  • #4
EvLer said:
shoot...i need sleep. :yuck:

Thanks :smile:

Good night, sleep tight! :zzz:

ehild
 
  • #5
sleep...highly recommended
 

What is a second order homogeneous differential equation with non-constant coefficients?

A second order homogeneous differential equation with non-constant coefficients is a mathematical equation that describes the relationship between a function and its derivatives up to the second order. It is considered homogeneous because all the terms in the equation have the same degree, and it has non-constant coefficients because the coefficients are not fixed values but can vary with respect to the independent variable.

What is the general form of a second order homogeneous differential equation with non-constant coefficients?

The general form of a second order homogeneous differential equation with non-constant coefficients is y'' + P(x)y' + Q(x)y = 0, where P(x) and Q(x) are functions of the independent variable x.

How is a second order homogeneous differential equation with non-constant coefficients solved?

A second order homogeneous differential equation with non-constant coefficients is solved by finding a particular solution that satisfies the equation. This can be done by using various methods such as the method of undetermined coefficients, variation of parameters, or the Laplace transform method.

What is the role of boundary conditions in solving a second order homogeneous differential equation with non-constant coefficients?

Boundary conditions are necessary in solving a second order homogeneous differential equation with non-constant coefficients because they provide additional information about the function and its derivatives at specific points. This allows for the determination of the particular solution that satisfies both the equation and the given conditions.

Can a second order homogeneous differential equation with non-constant coefficients have complex solutions?

Yes, a second order homogeneous differential equation with non-constant coefficients can have complex solutions. This occurs when the coefficients in the equation are complex numbers, and the solutions involve complex-valued functions. In this case, the real and imaginary parts of the solutions represent different physical phenomena.

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