Answer:
[tex]y(x)==(c_1+c_2x)e^{3x}+3x^2+4x[/tex]
Step-by-step explanation:
We are given that non-homogeneous equation
[tex]y''-6y'+9y=27x^2-18[/tex]
Particular solution of given equation is given by
[tex]y_p=3x^2+4x[/tex]
We have to find the general solution of the non-homogeneous equation.
Auxillary equation
[tex]m^2-6m+9=0[/tex]
[tex]m^2-3m-3m+9=0[/tex]
[tex]m(m-3)-3(m-3)=0[/tex]
[tex](m-3)(m-3)=0[/tex]
[tex]m=3,3[/tex]
[tex]y_c=(c_1+c_2x)e^{3x}[/tex]
General solution is given by
[tex]y(x)=y_c+y_p[/tex]
[tex]y(x)==(c_1+c_2x)e^{3x}+3x^2+4x[/tex]
