Answer to Question #304586 in Differential Equations for mou

Question

Solve The Differential Equation Y''+6y'+9y=16x^2

Answer 1

Corresponding homogeneous differential equation

$$y''+6y'+9y =16x^2$$

Characteristic (auxiliary) equation

$$r^2+6r+9=0$$

$$r_1=r_2=-3$$

The general solution of the homogeneous differential equation is

$$y_h=c_1e^{-3x}+c_2xe^{-3x}$$

Find the particular solution of the non homogeneous differential equation

$$y_p=Ax^2+Bx+C$$

$$y_p'=2Ax+B$$

$$y_p''=2A$$

Substitute

$$2A+12Ax+6B+9Ax^2+9Bx+9C=16x^2$$

$$A=16/9$$

$$B=-64/27$$

$$C=32/27$$

The general solution of the non homogeneous differential equation is

$$y=c_1e^{-3x}+c_2xe^{-3x}+\dfrac{16}{9}x^2-\dfrac{64}{27}x+\dfrac{32}{27}$$