Answer to Question #304586 in Differential Equations for mou

Question #304586

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

1
Expert's answer
2022-03-02T11:05:45-0500

Corresponding homogeneous differential equation


y+6y+9y=16x2y''+6y'+9y =16x^2

Characteristic (auxiliary) equation


r2+6r+9=0r^2+6r+9=0

r1=r2=3r_1=r_2=-3

The general solution of the homogeneous differential equation is


yh=c1e3x+c2xe3xy_h=c_1e^{-3x}+c_2xe^{-3x}

Find the particular solution of the non homogeneous differential equation


yp=Ax2+Bx+Cy_p=Ax^2+Bx+C

yp=2Ax+By_p'=2Ax+B

yp=2Ay_p''=2A

Substitute


2A+12Ax+6B+9Ax2+9Bx+9C=16x22A+12Ax+6B+9Ax^2+9Bx+9C=16x^2

A=16/9A=16/9

B=64/27B=-64/27

C=32/27C=32/27

The general solution of the non homogeneous differential equation is


y=c1e3x+c2xe3x+169x26427x+3227y=c_1e^{-3x}+c_2xe^{-3x}+\dfrac{16}{9}x^2-\dfrac{64}{27}x+\dfrac{32}{27}




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