The relationship between solutions

[ShayanJ]

Consider the linear ODE $y''-(ax-b)y=0$.
Wolframalpha gives $Ai(\frac{ax-b}{a^{2/3}})$ and $Bi(\frac{ax-b}{a^{2/3}})$ as the two independent solutions where Ai(z) and Bi(z) are the Airy functions.
I tried to find an answer myself, so I substituted $y=f(x) e^{cx}$ in the equation and tried to find f. The resulting ODE was hard so I gave it to wolframalpha and upon substituting what I got into y, I got:
$y=(\frac{ax-b}{c^2}-\frac{2a}{c^3})e^{cx}+k_1 x+k_2$
Now I'm confused about the relationship between these solutions. Because a 2nd order ODE has two independent solutions. So the solution I found should somehow be related to Airy functions which I fail to see how. Also it seems to me that there should be another solution independent of the solution I found so that I have two independent solutions too. Then it can be said that I just have two different bases for the space of the ODE's solutions. I doubt it though.
Any ideas?
Thanks

 

[JJacquelin]

Hi Shyan !
Put back your solution into the ODE and you will see that it doesn't agree. Your solution is false. The solution with the Airy functions is true. There is no relationship between a false and a true solution.