Question

d^2y/-6 dy/dx+9y=1+x+x^2

Answer 1

The homogeneous differential equation

$$\dfrac{d^2y}{dx^2}-6\dfrac{dx}{dy}+9y=0$$

The corresponding (auxiliary) equation

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

$$(r-3)^2=0$$

$$r_1=r_2=3$$

The general solution of the homogeneous differential equation

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

Find the particular solution of the nonhomogeneous differential equation

$$y_p=A+Bx+Cx^2$$

$$\dfrac{dy_p}{dx}=B+2Cx$$

$$\dfrac{d^2y_p}{dx^2}=2C$$

Substitute

$$2C-6(B+2Cx)+9(A+Bx+Cx^2)=1+x+x^2$$

$$9C=1$$ $$-12C+9B=1$$ $$2C-6B+9A=1$$

$$A=\dfrac{7}{27}, B=\dfrac{7}{27}, C=\dfrac{1}{9}$$

$$y_p=\dfrac{7}{27}+\dfrac{7}{27}x+\dfrac{1}{9}x^2$$

The general solution of the non homogeneous differential equation is

$$y=y_h+y_p$$

The solution of the given differential equation is

$$y=c_1e^{3x}+c_2xe^{3x}+\dfrac{7}{27}+\dfrac{7}{27}x+\dfrac{1}{9}x^2$$