Answer:
[tex]y_p(x) =c_1e^{-6x} + c_2e^{-3x}+ e^x + 4x^2[/tex]
Step-by-step explanation:
Given
[tex]y" + 9y' + 18y = 24x^2 + 40x + 8 + 12e^x[/tex] ---- (1)
[tex]y_p(x) = e^x + 4x^2[/tex]
Required
The general solution of [tex]y(x)[/tex]
Let
[tex]y = e^{nx}[/tex] be the trial solution of (1)
So:
[tex]y" + 9y' + 18y = 0[/tex] becomes
[tex]n^2 + 9n + 18 = 0[/tex]
Expand
[tex]n^2 + 6n+3n + 18 = 0[/tex]
Factorize
[tex]n(n + 6)+3(n + 6) = 0[/tex]
Factor out n + 6
[tex](n + 6)(n + 3) = 0[/tex]
Split
[tex]n +6 = 0\ or\ n + 3 = 0[/tex]
Solve for n
[tex]n =-6\ or\ n = -3[/tex]
So:
[tex]y = e^{nx}[/tex] becomes:
[tex]y = c_1e^{-6x} + c_2e^{-3x}[/tex]
[tex]y_p(x) = e^x + 4x^2[/tex] becomes
[tex]y_p(x) =c_1e^{-6x} + c_2e^{-3x}+ e^x + 4x^2[/tex]
Where: [tex]c_1[/tex] and [tex]c_2[/tex] are arbitary constants
