Help Solving Diff. Eq: y'' = xy

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Homework Help Overview

The discussion revolves around solving the differential equation y'' = xy, which falls under the subject area of differential equations.

Discussion Character

  • Exploratory, Conceptual clarification, Problem interpretation

Approaches and Questions Raised

  • Participants express uncertainty about how to approach the problem, with mentions of the Airy function as a potential solution. There are discussions about techniques such as separation of variables, Taylor series expansion, and the need for boundary conditions.

Discussion Status

The discussion includes various attempts to clarify methods for solving the equation, with some participants questioning the necessity of boundary conditions and others suggesting that arbitrary constants can be derived without them. There is a mix of exploration of different methods and expressions of confusion regarding the application of known techniques.

Contextual Notes

Some participants note the lack of boundary conditions and express uncertainty about their role in solving the equation, while others mention the challenge posed by the non-constant coefficient in the differential equation.

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Homework Statement


Solve the differential equation y'' = xy


Homework Equations


-


The Attempt at a Solution


Not too sure what to do here. I looked it up and apparently the solution to this is the Airy function, but I don't understand how they got there. Help would be appreciated.
 
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FeDeX_LaTeX said:

Homework Statement


Solve the differential equation y'' = xy


Homework Equations


-


The Attempt at a Solution


Not too sure what to do here. I looked it up and apparently the solution to this is the Airy function, but I don't understand how they got there. Help would be appreciated.

What techniques do you know for solving differential equations? I have a particular method in mind, but there are possibly others.
 
Separation of variables and solving 1st and 2nd order DEs with constant co-efficients -- not sure what to do here though since the co-efficient of y isn't a constant. I've heard that you solve this with a Taylor series expansion but I don't know how to apply that here. I've studied doing it by finding roots of the auxiliary equation and Taylor series, but I'm unsure of how to use the Taylor series method for this equation.
 
Last edited:
You need boundary conditions to solve by method of Taylor series but I can't remember what they were. How would I approach this problem without knowing the boundary conditions (for the general case)?
 
FeDeX_LaTeX said:
You need boundary conditions to solve by method of Taylor series but I can't remember what they were. How would I approach this problem without knowing the boundary conditions (for the general case)?

You don't need boundary conditions. You will discover that you have no values for ##a_0## and ##a_1## and can get all the ##a##'s in terms of them. They will wind up being the two arbitrary constants.
 
Never mind, solved it.
 

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