✔ 最佳答案
y" - 2y' + 5y = 25x2 + 12
Auxiliary equation:
k2 - 2k + 5 = 0
k = 1 +- 2i
So, the complementary solution:
yc = ex(Asin2x + Bcos2x)
Let particular solution, yp = Cx2 + Dx + E
yp' = 2Cx + D
yp" = 2C
(2C) - 2(2Cx + D) + 5(Cx2 + Dx + E) = 25x2 + 12
5Cx2 + (5D - 4C)x + (5E - 2D + 2C) = 25x2 + 12
Comparing coefficients,
5C = 25, C = 5
5D - 4(5) = 0
D = 4
5E - 2(4) + 2(5) = 12
E = 2
So, the general solution, y
= yc + yp
= ex(Asin2x + Bcos2x) + 5x2 + 4x + 2, where A and B are constants
2009-08-21 14:21:33 補充：
This is the general solution.
General solution = Complementary solution + Particular solution.
If you want to determine the value of A and B, two initial conditions should be given.
2009-08-21 14:27:28 補充：
This is the general solution, not particular solution.
Just this is a non-homongenous 2nd diff. eqt, so we need to add complementary solution and particular solution to get the general solution.
