Method of undetermined coefficent

[vabamyyr]

method of undetermined coefficent is applied when g(x) is exponent, polynomial or trigonometric function. What will happen if g(x)= lnx or some other log function
ay`` + by`+ cy = g(x), where g(x)=log function. What method to use then?

 

[dextercioby]

If "a","b","c" are constants wrt "x",then Lagrange's method cannot fail.The method of variation of constants.

Daniel.

 

[M98Ranger]

If g(x) is a logarithmic function, the method of undetermined coefficients cannot be directly applied. Instead, the method of variation of parameters can be used to find a particular solution. This method involves finding a particular solution in the form of y_p=u(x)y_1+v(x)y_2, where y_1 and y_2 are the two linearly independent solutions of the homogeneous equation (ay`` + by`+ cy=0) and u(x) and v(x) are functions to be determined. The solution can then be substituted back into the original equation to solve for u(x) and v(x). This method is more complex than the method of undetermined coefficients, but it can be used for any type of non-homogeneous function, including logarithmic functions.