# Solve y'' = xy

Gold Member

## Homework Statement

Solve the differential equation y'' = xy

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## 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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Mark44
Mentor

## Homework Statement

Solve the differential equation y'' = xy

-

## 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.

Gold Member
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.

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Gold Member
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)?

LCKurtz
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.