Solving O.D.E. Given Sequence (c, s, 1, e_1, e_{-1})

In summary, the conversation discusses finding the differential equation with the solution space given by the basis (c, s, 1, e_1, e_{-1}), where c = cos(t) and s = sin(t). The discussion also mentions e_1 and e_{-1} as potential values for the basis and explains the concept of Ker(x). The expert provides a possible solution using the roots of the characteristic equation i, -i, 0, 1, and -1, but points out that there may be multiple solutions for the given problem.
  • #1
VinnyCee
489
0
Problem:

The sequence (c, s, 1, [tex]e_1,\,e_{-1}[/tex]) is a basis for the solution space of some differential equation p(D)y = 0. Find this O.D.E.

NOTE: c = cos(t) and s = sin(t)


Work so far:

I know that [tex]e_1[/tex] gives a (t - 1) and that the [tex]e_{-1}[/tex] gives a (t + 1), but how do I solve for the 1! I think that the c and s give ([tex]t^2[/tex] - 1).

Also, can someone explain in detail or give a reference to what a Ker() is?

thanks
 
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  • #2
Ker(x) means the kernel of the transformation x--the set of all values that x maps to the identity. In linear algebra that would be the set of all values that x maps to the zero vector. What exactly are [tex]e_1,\,e_{-1}[/tex]?
 
  • #3
VinnyCee said:
Problem:

The sequence (c, s, 1, [tex]e_1,\,e_{-1}[/tex]) is a basis for the solution space of some differential equation p(D)y = 0. Find this O.D.E.

NOTE: c = cos(t) and s = sin(t)


Work so far:

I know that [tex]e_1[/tex] gives a (t - 1) and that the [tex]e_{-1}[/tex] gives a (t + 1), but how do I solve for the 1! I think that the c and s give ([tex]t^2[/tex] - 1).
I have no idea what you mean by this! Is it possible that [tex]e_1[/tex] and [tex]e_{-1}[/tex] were supposed to be [itex]e^1[/itex] and [tex]e^{-1}[/itex]? If that is the case then the roots of the characteristic equation are i, -i, 0, 1, and -1. From that information, you should be able to find the characteristic equation and from that the differential equation.

The problem as given ("Find this O.D.E.") has no single solution. There exist an infinite number of differential equations having those functions as solutions. Why I am giving is the simplest linear, homogenous, differential equation.
 
  • #4
I think he meant [tex]e^t[/tex] and [tex]e^{-t}[/tex] since c, s, and 1 are functions (well, e and 1/e would be functions too, but for these purposes would be equivalent to 1)
 

1. What is an O.D.E?

An O.D.E (Ordinary Differential Equation) is a mathematical equation that describes the relationship between a function and its derivatives. It is used to model many real-world phenomena in fields such as physics, engineering, and biology.

2. What is the purpose of solving an O.D.E given a sequence?

The purpose of solving an O.D.E given a sequence is to find an explicit solution to the equation, which can then be used to make predictions and analyze the behavior of the system described by the O.D.E.

3. How do you solve an O.D.E given a sequence?

To solve an O.D.E given a sequence, you can use various methods such as separation of variables, integrating factors, or the method of undetermined coefficients. The specific method used will depend on the type of O.D.E and the initial conditions given.

4. What is the role of the constants c, s, 1, e_1, e_{-1} in solving an O.D.E given a sequence?

The constant c represents the initial condition of the function, while s represents the initial condition of the derivative of the function. The constants 1, e_1, and e_{-1} are used in specific methods of solving O.D.Es, such as the method of undetermined coefficients.

5. Can an O.D.E given a sequence have multiple solutions?

Yes, an O.D.E given a sequence can have multiple solutions. This is because the initial conditions given may result in different solutions to the equation. In some cases, an O.D.E may also have an infinite number of solutions.

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