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.

 

[tacman]

Hi AB,

The D in this equation represents the derivative operator, which is a mathematical notation used to represent the derivative of a function. In this case, the equation is a third-order ordinary differential equation (ODE), which means it involves the third derivative of the function y.

When solving ODEs, it is important to properly identify and understand the notation being used. In this case, the D represents the derivative operator, and the equation is asking you to find a function y that satisfies the equation when its third derivative is multiplied by X and added to its second derivative multiplied by 4.

I am glad to hear that you were able to solve some of the problems using the undetermined coefficient method. This is a useful technique for finding the particular solution of an ODE when the general solution of the homogeneous equation is already known.

I would suggest reviewing the concept of derivative operators and their use in ODEs to further understand this type of equation. Good luck with your homework!