Find Closed Form of Differential Equation: y''' - y = 0

In summary, the given power series is a solution to the differential equation y''' - y = 0. By using the characteristic polynomial, it can be determined that the closed form of the series is likely C*e^x, though this is only one possible solution and all solutions must be found.
  • #1
fittipaldi
3
0
Hi, everyone, I need some help with the following:

Homework Statement



Given is, that the following power series:
[tex]\sum_{n=0}^{\infty} \frac{x^{3n}}{(3n)!}[/tex]

is the solution to the following differential equation: y''' - y = 0. Find the closed form of the series.

Homework Equations



None

The Attempt at a Solution



Well I tried differentiating the sum, then applying it to the differential equation, but I get something very nasty, so I am sure, I am on a bad way.

Please, help!
 
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  • #2


fittipaldi said:
Hi, everyone, I need some help with the following:

Homework Statement



Given is, that the following power series:
[tex]\sum_{n=0}^{\infty} \frac{x^{3n}}{(3n)!}[/tex]

is the solution to the following differential equation: y''' - y = 0. Find the closed form of the series.

Homework Equations



None

The Attempt at a Solution



Well I tried differentiating the sum, then applying it to the differential equation, but I get something very nasty, so I am sure, I am on a bad way.

Please, help!

Well, you're given that that particular series is a solution to y''' - y = 0. You can also solve this another way. Might I recommend characteristic polynomials?

(also, doesn't that power series look suspiciously close to the power series for e^x?)
 
  • #3


Char. Limit said:
Well, you're given that that particular series is a solution to y''' - y = 0.

That is true, but I cannot seem to understand how to continue ... a bigger tip maybe?

Char. Limit said:
(also, doesn't that power series look suspiciously close to the power series for e^x?)

Also noticed that, the only difference is the coefficient. C*e^x is also one of the solutions to the given equation, how does this help?
 
  • #4


fittipaldi said:
That is true, but I cannot seem to understand how to continue ... a bigger tip maybe?
Also noticed that, the only difference is the coefficient. C*e^x is also one of the solutions to the given equation, how does this help?

Well, it's entirely possible that the closed form of your power series is C*e^x for some C. In fact, it looks like that's the case to me.

EDIT: After checking, I can conclude that C*e^x is PART of the solution. However, you'll need to find ALL solutions to the differential equation. C*e^x is just one.
 
  • #5
hi fittipaldi! :smile:
fittipaldi said:
That is true, but I cannot seem to understand how to continue ... a bigger tip maybe?

as Char. Limit :smile: says, use the characteristic polynomial :wink:
 

1. What is a "closed form" of a differential equation?

A closed form of a differential equation is an equation that can be expressed in terms of elementary functions, such as polynomials, trigonometric functions, and exponential functions. It does not include any integrals, derivatives, or other operations that cannot be easily solved algebraically.

2. How do you find the closed form of a differential equation?

To find the closed form of a differential equation, you need to solve for the dependent variable (usually denoted as y) in terms of the independent variable (usually denoted as x). This can be done by using techniques such as separation of variables, substitution, or integrating factors.

3. What does the notation y''' - y = 0 mean in a differential equation?

The notation y''' - y = 0 indicates a third-order differential equation, where the third derivative of y (y''') is equal to the function y itself. This type of equation is also known as a homogeneous linear differential equation.

4. Why is finding the closed form of a differential equation important?

Finding the closed form of a differential equation allows us to find a specific solution that satisfies the equation for all values of the independent variable. This can help us understand the behavior of a system over time and make predictions about its future behavior.

5. Can all differential equations be solved to find a closed form?

No, not all differential equations can be solved to find a closed form. Some equations are non-linear or non-homogeneous, making them more difficult or impossible to solve using traditional methods. In these cases, numerical methods or approximations may be used to find an approximate solution.

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