1. ## Particular Integral

Hello, I was just wondering if there is a particular integral for a second-order linear inhomogeneous differential equation which is equal to lnx.

I mean we are taught at university to use some particular integrals for some cases eg when we meet sinx, cosx, a polynomial, e^x but no one told us what to do if we find cotx, tanx, lnx, logx or any other function except of the three-four cases which are standard. Any reference for a complete list of particular integrals would be much appreciated!

2. The standard definition of the natural logarithm is $\ln{x} = \int_{1}^{x}\frac{1}{t}\;{dt}.$
Is that what you were looking for? Or do you want to integrate $\ln{x}$?

3. No, my question is what do we use as a particular integral when a second order linear inhomogeneous differential equation has as a right side the term lnx.
I mean when the right side is e^x we use Ae^x, when it is x, we use Ax+b. If that term is lnx or tanx or cotx eg what do we use as a particular integral?

4. I don't think the guess-and-check method will work here. You'll have to do variation of parameters to solve problems like that. Warning: logarithms are not liable to give you pretty results. Here's an example.

5. Thank you very much Ackbeet. I am first year student and I am pretty interested in Differential Equations, that is why I asked. I think you know that in first year calculus we do not meet any other method except the particular integral! Thanks again for the information!

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# particular integral of logx*sin(logx)

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