What Does 'D' Stand for in ODEs?

[brunie]

Hi,
I was doing a few homework questions and I came about a few ODEs of this form.
For example,

(XD^3 + 4D^2)y = 8e^x

Im not sure what the D stands for, is it a constant, is it a derivative of y (ie. y'''), etc. ??

In the same section of problems there were questions that I solved using undetermined coefficient method by finding both the homogenous and particular solutions.

Any help would be appreicated.
AB

 

[HallsofIvy]

"D" represents the differential operator. D^2 is the second derivative and D^3 is the third derivative. Is the X in "XD^3" supposed to be x?

If you let u= D^2 y (i.e. u= y") then that equation reduces to the linear first order equation xu'+ 4y= 8e^x. You can solve that by finding the integrating factor and then integrate twice more to find y.