Find a particular solution of the nonhomogeneous equation and give the general solution of the equation.

y''-(1/x)y'+(1/x^2)y=2/x

y1(x) = x and y2(x) = xlnx

Question

Find a particular solution of the nonhomogeneous equation and give the general solution of the equation.

y''-(1/x)y'+(1/x^2)y=2/x 

y1(x) = x and y2(x) = xlnx

anonymous · Answer

For a solution, i got z(x) = 2lnx + 2lnx(xlnx)
but that seems like it would be wrong.

So I doubt my general solution is correct either.

Can someone please check this?

experimentx · Answer

this looks like Euler-Cauchy equation

anonymous · Answer

it doesnt really mention that in the book... it just kind of gives guidelines how to find the solutions.  they use the wronskian and the equation z(x) = u(x) y1+v(x) y2 to find the solution and general solution.

experimentx · Answer

http://en.wikipedia.org/wiki/Cauchy%E2%80%93Euler_equation

experimentx · Answer

try substituting z=ln(x) ... you will have second order ODE with e^z in terms of y and z ... solving it quite easy ... just substitute back x from z

anonymous · Answer

you mean for my solution?

experimentx · Answer

no i haven't check it ... what method did you use to deduct it?

experimentx · Answer

According to Mathematica, the solution is $ x C[1]+x C[2] \text{Log}[x]+x \text{Log}[x]^2 $

anonymous · Answer

well i calculated the wronskian first using my y1 and y2 and got W(x) = x

then I calculated u(x) = integral (-y2)f(x))/Wx

then I calculated v(x) = inegral (y1/f(x)/Wx

then plugged into z(x) = u(x) y1 + v(x) y2

experimentx · Answer

I mean what method did you use to deduct y1(x) = x and y2(x) = xlnx ??

anonymous · Answer

oh, i didnt, it was given.

anonymous · Answer

we are just told to verify they are solutions

experimentx · Answer

the method it's asking is called variation of parameters http://en.wikipedia.org/wiki/Variation_of_parameters

anonymous · Answer

yes... that is it.

experimentx · Answer

http://upload.wikimedia.org/math/9/4/3/943ea8e2599a7ef999367efdf5b5fac5.png

experimentx · Answer

in this picture, W is your wonskian, u1 and u2 are your given homegeneous equation's solution ... and put f(x)=1/x and evaluate those integrals.

experimentx · Answer

woops!! wrong picture http://upload.wikimedia.org/math/1/5/7/157873c4bc16e492763bda5edd135911.png

experimentx · Answer

and add them ... A(x) and B(x) ... and don't forget to add integration of constants.

anonymous · Answer

okay now I got z(x) = x(ln x)^2

experimentx · Answer

all right ... that's correct. according to W|A

anonymous · Answer

ohhh, well what is W|A, or whatever that was that you used?

experimentx · Answer

http://www.wolframalpha.com/input/?i=y%27%27-%281%2Fx%29y%27%2B%281%2Fx%5E2%29y%3D2%2Fx+

anonymous · Answer

ohhh thats pretty cool... i have never tried that site... well great way to check my answers for sure.. thank you a bunch

experimentx · Answer

yw