# Help in solving a second-order linear differential equation

$$\frac{{d^2 y}}{{dx^2 }} + \left( {Ax + B} \right)y = 0$$

I have tried lots of substitions, but a solution won't pop out. Can anyone help solve this?

Thanks.

Make the change of variable

$$Ax + B = \lambda u$$

(lambda is a constant) this will give you

$$\frac{A^2}{\lambda^2} \frac{d^2 y}{d u^2} + \lambda u y = 0$$

so if you then set

$$\lambda = -A^{\frac{2}{3}}$$

you then have

$$\frac{d^2 y}{d u^2} - u y = 0$$

which is the Airy equation (in u). Have a look on Wikipedia or elsewhere on Airy functions and such - or just type in "Airy Equation".

edit: note you will actually get three different solutions as the condition for lambda is

$$\frac{\lambda^3}{A^2} = -1$$

which means that there are three values of lambda that satisfy this (i.e. three distinct cube-roots) - one will be real-valued (already given) plus two complex ones.

Here is a link for the Airy function

http://mathworld.wolfram.com/AiryFunctions.html

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