Nonhomogeneous Second Order ODE containing log

[crowy]

Hi guy,

I have this ODE that I'm having problems with

y"+4y'+4y= e^(-2x)logx

Now, Using method of UC to get rid of the RHS I've tried using Ae^(-2x) x^2 logx

However, I'm not quite sure whether that is correct or not as I have never had a question containing logs before

 

[ross_tang]

If it is not a must that you have to use method of undetermined coefficients, you can have a look of the operator method. In this case, you don't have to care about what kind of non-homogeneous function you have. At least you can write the solution in integral form. Please refer to my tutorial in http://www.voofie.com/concept/Mathematics/" :

http://www.voofie.com/content/6/introduction-to-differential-equation-and-solving-linear-differential-equations-using-operator-metho/"

For equations with variable coefficients, you can at least try to find the solution using the below method:

http://www.voofie.com/content/84/solving-linear-non-homogeneous-ordinary-differential-equation-with-variable-coefficients-with-operat/"

If you want to see how to solve it step by step, you can try to ask it in http://www.voofie.com/concept/Mathematics/" by submitting a new question, and I am willing to solve it for you.

 

[JJacquelin]

See the attachement :

 

[HallsofIvy]

In general, the "method of undetermined coefficients" can only be used when the right hand side is the type of function that might be a solution to a homogeneous linear equation with constant coefficients- sine and cosine, polynomials, exponentials, and combinations of those.

The method of "variation of parameters" works with any functions- although it may result in integrals the require non-elementary functions.