Unsolvable Equation: Solve for y

[superg33k]

Homework Statement

Solve for y: (note the one in the title is wrong)

$$\frac{d^2y}{dx^2} = -2xy \frac{dy}{dx}$$

Homework Equations

None that I can think of.

The Attempt at a Solution

[strike]Oh so many failures. None which solved anything. I don't even know if this equation is solvable. :([/strike]

Well I think the correct solution should contain two unknowns. Two solutions are y=A and y=3/2 x^(-2) but a linear combination of these is not. Also no other solutions can be expressed as a finite polynomial. So where the heck do I go from here?

 

[mighty2000]

Thank you for bringing this equation to our attention. I can assure you that this equation is indeed solvable. The equation you have provided is a second-order differential equation, and there are various methods for solving such equations.

One approach is to use the method of variation of parameters, where we assume the solution to be in the form of y = u(x)y1(x) + v(x)y2(x), where y1 and y2 are two linearly independent solutions to the homogeneous equation (i.e. when the right-hand side is equal to 0). We can then solve for u(x) and v(x) using the given boundary conditions.

Another approach is to use power series methods, where we assume the solution to be in the form of a power series and then solve for the coefficients using the given boundary conditions.

There are also other methods such as Laplace transforms and the method of undetermined coefficients that can be used to solve this type of equation.

I hope this helps. If you need further assistance, please don't hesitate to ask. As scientists, it is our job to solve complex equations and problems, so don't be discouraged by initial failures. Keep trying and you will find the solution.