Xy''−xf(x)y' + f(x)y = 0?

<img src="https://tex.z-dn.net/?f=%5Ctexttt%7BSea%20%7Dx%5Cneq%200%5Ctexttt%7B%20entonces%20hacemos%3A%20%7D%5C%5C%20%5C%5C%0A%5Chspace%2A%7B3cm%7Dy%27%27-f%28x%29y%27%2B%5Cdfrac%7Bf%28x%29%7D%7Bx%7Dy%3D0%5C%5C%20%5C%5C%20%5C%5C%0A%5Ctexttt%7BLuego%20un%20cambio%20de%20variable%3A%20%7Du%28x%29%3D%5Cdfrac%7By%7D%7Bx%7D%5Cto%20y%3Dux%5C%5C%20%5C%5C%20%5C%5C%0A%28ux%29%27%27-f%28x%29%28ux%29%27%2B%5Cdfrac%7Bf%28x%29%7D%7Bx%7D%28ux%29%3D0%5C%5C%20%5C%5C%20%5C%5C%0A%28u%27%27x%2B2u%27%29-f%28x%29%28u%27x%2Bu-u%29%3D0%5C%5C%20%5C%5C%20%5C%5C%0Au%27%27x%2B%5Cleft%5B2-xf%28x%29%5Cright%5Du%27%3D0%5C%5C%20%5C%5C%20%5C%5C%0Au%27%27%2B%5Cdfrac%7B2-xf%28x%29%7D%7Bx%7Du%27%3D0%5C%5C%20%5C%5C%20%5C%5C%0A%5Ctexttt%7BSea%20%7Du%27%3Dw%28x%29%5Ctexttt%7B%20entonces%3A%20%7D%5C%5C%20%5C%5C%20%5C%5C%0Aw%27%2B%5Cleft%28%5Cdfrac%7B2%7D%7Bx%7D-f%28x%29%5Cright%29w%3D0%0A%0A%0A" />

<img src="https://tex.z-dn.net/?f=%5Cdfrac%7Bdw%7D%7Bdx%7D%3D%5Cleft%28f%28x%29-%5Cdfrac%7B2%7D%7Bx%7D%5Cright%29w%5C%5C%20%5C%5C%20%5C%5C%0A%5Cdfrac%7Bdw%7D%7Bw%7D%3D%5Cleft%28f%28x%29-%5Cdfrac%7B2%7D%7Bx%7D%5Cright%29dx%5C%5C%20%5C%5C%20%5C%5C%0A%5Cdisplaystyle%0A%5Cln%20%7Cw%7C%20%3D%5Cint%20f%28x%29-%5Cdfrac%7B2%7D%7Bx%7D%5C%3B%20dx%2BC_1%5C%5C%20%5C%5C%20%5C%5C%0A%5Cln%20%7Cw%7C%20%3D%5Cint%20f%28x%29%5C%2Cdx-2%5Cln%20x%2BC_1%5C%5C%20%5C%5C%20%5C%5C%0Aw%3D%5Cdfrac%7BC_2%5Ccdot%20e%5E%7B%5Cint%20f%28x%29%5C%2Cdx%7D%7D%7Bx%5E2%7D%5C%5C%20%5C%5C%20%5C%5C%0Au%3DC_2%5Cint%20%5Cdfrac%7Be%5E%7B%5Cint%20f%28x%29%5C%2Cdx%7D%7D%7Bx%5E2%7D%5C%2C%20dx%2BC_3%5C%5C%20%5C%5C%20%5C%5C%0A%5Cboxed%7By%3Dx%5Ccdot%20C_2%5Cint%20%5Cdfrac%7Be%5E%7B%5Cint%20f%28x%29%5C%2Cdx%7D%7D%7Bx%5E2%7D%5C%2C%20dx%2Bx%5Ccdot%20C_3%7D" /> <img src="https://tex.z-dn.net/?f=%5Ctexttt%7BDonde%20los%20%7DC_i%5Ctexttt%7B%20son%20constantes%20y%20%7D%20%5Cint%20f%28x%29%5C%2Cdx%20%5Ctexttt%7B%20se%20considera%20como%7D%5C%5C%20%0A%5Ctexttt%7Buna%20funci%5C%27on%20de%20solo%20la%20variable%20%7Dx%0A" />.