Differential Equation - Hermite's Equation

In summary, the general solution for the given differential equation is y = e^(-t^2) + C. This is obtained by dividing the equation by y and integrating.
  • #1
cse63146
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0

Homework Statement



Find the general solution of y' + 2ty = 0

Homework Equations





The Attempt at a Solution



so I know y = [tex]\Sigma^{\infty}_{n=0} a_n t^n[/tex] and y' =[tex]\Sigma^{\infty}_{n=1} n a_n t^{n-1}[/tex]

not sure what to do next.
 
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  • #2
cse63146 said:

Homework Statement



Find the general solution of y' + 2ty = 0

Homework Equations





The Attempt at a Solution



so I know y = [tex]\Sigma^{\infty}_{n=0} a_n t^n[/tex] and y' =[tex]\Sigma^{\infty}_{n=1} n a_n t^{n-1}[/tex]

not sure what to do next.

http://www.sosmath.com/diffeq/series/series06/series06.html" [Broken]
 
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  • #3
Is that first line a typo? Should it be y'' (two primes, second derivative)? I'm guessing it is because the link shows a second order DE.

If it is a single prime, then you could do a simple integration to get the answer, unless you have been told to use a series solution.
 
  • #4
It's y' + 2ty = 0. Makes the problem a bit easier.
 
  • #5
Yah wait a minute, what does that equation have to do with Hermite's equation? Are you SURE it isn't a second order differential equation?
 
  • #6
Find the general solution of y' + 2ty = 0
Do you have a prescribed method (like series?)
If not, it is very simple
Devide the Eq by y and you get
y´/y = -2t
Integrate
ln(y) = -t2
y=exp(-t2) + C
that is the general solution
 
  • #7
It might not be (even though its in the same section). But I wanted to try and solve that before I move on to the second order version.
 

1. What is Hermite's equation?

Hermite's equation is a type of differential equation that is used to model the behavior of a physical system. It is named after the French mathematician Charles Hermite, who first studied this type of equation in the 19th century.

2. How is Hermite's equation different from other types of differential equations?

Hermite's equation is a special type of differential equation known as a second-order linear ordinary differential equation. It is characterized by the presence of a polynomial function with a variable coefficient, which is what makes it unique from other types of differential equations.

3. What are some real-world applications of Hermite's equation?

Hermite's equation is commonly used in physics and engineering to model the behavior of systems such as oscillators, electric circuits, and quantum harmonic oscillators. It is also used in statistics to describe the probability distribution of a random variable.

4. How do you solve Hermite's equation?

Solving Hermite's equation involves finding a solution to the differential equation that satisfies the given initial conditions. This can be done using various methods such as power series, Laplace transforms, or numerical methods. The specific method used will depend on the complexity of the equation and the initial conditions.

5. What are the properties of solutions to Hermite's equation?

Solutions to Hermite's equation have some distinct properties, such as being infinitely differentiable and having a specific form involving Hermite polynomials. They are also oscillatory in nature and can have complex-valued solutions. Additionally, solutions to Hermite's equation can be used to describe the behavior of a system over time and can provide insights into its stability and convergence.

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