Here's the question you clicked on:
RolyPoly
Solve by undermined coefficients/variation of parameters: \[ y''-4y'+4y = \frac{e^{2x}}{x}\]
\[ y_c''-4y_c'+4y_c=0\]\[ \lambda^2-4\lambda+4=0\]\[ \lambda=2, 2\]\[y_c = c_1e^{2x}+c_2xe^{2x}\] How can I guess the particular solution?
you can't do this one with undetermined cofficients. you would have to use variation of parameter.
Why doesn't undetermined coefficients work in this case??
well in undetermined cofficients there are few general forms of the solution already told.. i mean like if f(x) is e^x = Ae^ax.. and a few more like these.. know of it?
for "x" general form of its solution of yp would be Ax+B
Is xe^(2x) a general form?
general for have unknown cofficients.. this one is not general form.. but this has a general for , it would be (Ax+B)e^(cx)
Then how do you define ''general''??
like for equation of a line y=mx+c.. this is general form. u ahve to find x and m.. same way in here yp = (Ax+B)e^(cx) is general form and we have to find A,B,c. th
Some typical examples are constant - constant sin / cos - Asinx + Bcosx e^(ax) = Ce^(ax) x^n = polynomial of x But xe^(2x) is not that ''general''
in exams we never get the straight one .. in here we have to first see if x has general for or not.. then we would see if e^2x has general form or nto.. if they both would have just miltiply them
\(x e^{2x}\) still qualifies as general
and e^(2x) / x on the other hand is not general?
in my opinion, No. because i think that x in the denominator part is the trouble here and we dont have a general form for that part. so i think variation of parameter should do the trick.
Variation of parameters: \[y'' - 4y' + 4y = \frac{e^{2x}}{x}\]\[y_c'' - 4y_c' + 4y_c =0\]\[y_c = c_1e^{2x}+c_2xe^{2x}\] \[u_1'e^{2x} + u_2' xe^{2x}=0\]\[2u_1'e^{2x} + u_2' (2xe^{2x}+e^{2x})=\frac{e^{2x}}{x}\] \[u_2'e^{2x}=\frac{e^{2x}}{x}\]\[u_2 = lnx\] \[u_1' = -\frac{\frac{1}{x}xe^{2x}}{e^2x}\]\[u_1 = -x\] \[y=c_1e^{2x}+c_2xe^{2x} +xlnxe^{2x} - xe^{2x}\] I can't believe that I got the solution! :O
lol well i am glad it worked out for u :)