y_p:
_______________________________
y = cx3e^{-x} + ax + b
y' = e-x [aex -cx3 +3cx2]
y'' = e-x [x2-6x+6] cx
y'''= -c e-x [x3-9x2+18x-6]
____________
y'''+3y''+3y'+y=x+6e-x
x+6e-x =e-x [-c(x3-9x2+18x-6)] + e-x [3cx(x2-6x+6)] + e-x [3(aex -cx3 +3cx2)] +
e-x [cx3] + ax + b
= ax + b + e-x[-c(x3-9x2+18x-6)+3cx(x2-6x+6)+3(aex -cx3 +3cx2)+cx3]
= ax + b + e-x[3(a*ex+2c)]
= ax + 3a + b + 6ce-x
=> c = 1
a = 1
3a+b = 0 => b = -3
y_p = cx3 e-x + ax + b
= x3 e-x + x -3
_______________________________
y = y_h + y_p =
(ax2+bx+c) e^{-x} + x^3e^{-x} + x -3 =
e^{-x} [ax2+bx+c + x^3
] + x -3 =
e^{-x} [x^3+
ax2+bx+c] + x -3
___sieht das alles so richtig aus?
MFG