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

Answer:

$y(x)==(c_1+c_2x)e^{3x}+3x^2+4x$

Step-by-step 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

$y(x)=y_c+y_p$

$y(x)==(c_1+c_2x)e^{3x}+3x^2+4x$