What Does 'D' Stand for in ODEs?

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SUMMARY

The letter 'D' in ordinary differential equations (ODEs) represents the differential operator, where D^2 denotes the second derivative and D^3 denotes the third derivative of a function y. In the example provided, the equation (XD^3 + 4D^2)y = 8e^x can be transformed by letting u = D^2y, simplifying it to a linear first-order equation xu' + 4y = 8e^x. This can be solved using the integrating factor method followed by integration to find the solution for y.

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brunie
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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
 
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"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.
 

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