$y=x^r$ kabul et ve turevlerini yerine koyarsan, cift kok geldiginden homojen cozum su olur:
$y_h=c_1x^2+c_2x^2lnx$
Parametrelerin degisimi yontemini kullanirsak:
$y_1=x^2$ $y_2=x^2lnx$ olsun.
Wronskian: $W=y_1y_2'-y_2y_1'\neq0$
$W=x^2(2xlnx+x^2/x)-x^2lnx2x=x^3\neq0$
Ozel cozum soyle bulunur: $$y_o=y_2\int\frac{y_1f(x)}{W}dx-y_1\int\frac{y_2f(x)}{W}dx$$
$$y_o=x^2lnx\int\frac{x^2(ln^2x+1)}{x^3}dx-x^2\int\frac{x^2lnx(ln^2x+1)}{x^3}dx$$
$$y_o=x^2lnx\int\frac{(ln^2x+1)}{x}dx-x^2\int\frac{lnx(ln^2x+1)}{x}dx$$
$$y_o=x^2lnx\left(\int\frac{ln^2x}{x}dx+\int\frac{1}{x}dx\right)-x^2\left(\int\frac{ln^3x}{x}dx+\int\frac{lnx}{x}dx\right)$$
$lnx=u$ donusumu yaparsan ozel cozumu bulursun. Genel cozum ise homojen + ozel cozum olur..
http://www.acikders.org.tr/file.php/4/LectureNotesAndReadings/D13.pdf
http://tutorial.math.lamar.edu/Classes/DE/VariationofParameters.aspx