حل معادله تفاضليه صعبه

y** +9y=5sin2x+4 solve
the Solution

المعادلة البديلة

or auxiliary equation is

y2 + 9 = 0‎
y = ± 3i then a = 0 , abs ( b) = 3 ‎

yh = a sin3x + b cos3x : a, b are constants

particular solution : yp

yp = a sin2x + b cos2x + c : a, b ,c constants
yp\ = 2a cos2x – 2b sin 2x
yp\\ = – 4a sin2x – 4 b cos2x ...... 1‎
‎9yp = 9a sin2x + 9b cos2x + 9c ......2‎
yp\\ + 9 yp = 5a sin2x + 5b cos2x + c = 5 sin2x + 4‎
then : a = 1 , b = 0 , c = 4/9
‎
yp = sin2x + 4/9
‎
y = yh + yp ( مبدأ التراكم)

y = a sin3x + b cos3x + sin2x + 4/9‎

التوقيع :