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in undetermined coefficients when r(x) has sum of two unlike function

like

e^ax+x^n,we find yp(particular solution) using yp=yp1+yp2(yp1 for e^ax and yp2 for x^n)

but for like functions ,is

sinx+cosx or x^ax+x^3x

we have to find yp using sum of yp1 and yp2 or we find only yp using sin(neglecting cos) or x^ax(neglecting x^3x)?

I assume that, in your question, r(x) is the inhomogeneous term. I also am a little concerned about your function x^ax, which is unusual and does not lend itself to solution by the method of undetermined coefficients. I'll assume that you mean something like x^n + x^m, where n and m are integers.

For the trig functions, if you have just a single trig function (not the sum), i.e., r(x) = Asin(ax) or r(x) = Bcos(bx), you're going tho have to use both the sin and cos functions in your particular solution anyway since, in plugging them in to the differential equation and matching coefficients, you are going to have terms like d[sin(ax)]/dx = acos(ax), so you may as well include both sin and cos in your particular solution from the start.

In the general case for the trig functions, if you have the sum of trig functions with different coefficients in the argument, i.e., r(x) = Asin(ax)+Bcos(bx), then you should split the problem up by using the superposition (i.e., the sum) of the solutions for each case (sin(bx) and/or cos(bx)) separately.

For polynomials, a similar situation occurs, namely, when r(x)=x^n and a term like x^n is used in the particular solution, the derivatives will give d(x^n)/dx = nx^(n-1), so term of lower order will be necessary in the particular solution.

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