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

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