Question

The function yp=3x2+4x is a particular solution to the nonhomogeneous equation y′′−6y′+9y=27x2−18 Find the general solution of the nonhomogeneous equation y′′−6y′+9y=27x2−18. (Hint: you need yc.)

Answer 1

`y(x)==(c_1+c_2x)e^(3x)+3x^2+4x`

Explanation:

We are given that non-homogeneous equation

`y''-6y'+9y=27x^2-18`

Particular solution of given equation is given by

`y_p=3x^2+4x`

We have to find the general solution of the non-homogeneous equation.

Auxillary equation

`m^2-6m+9=0`

`m^2-3m-3m+9=0`

`m(m-3)-3(m-3)=0`

`(m-3)(m-3)=0`

`m=3,3`

`y_c=(c_1+c_2x)e^(3x)`

General solution is given by

`y(x)=y_c+y_p`

`y(x)==(c_1+c_2x)e^(3x)+3x^2+4x`