Question

Find the general solution of the following differential equations using method of undetermined coefficients: (i) y''−2y'+y =x2e5x,

Answer 1

Corresponding homogeneous differential equation

$$y''−2y'+y =0$$

Characteristic (auxiliary) equation

$$r^2-2r+1=0$$

$$r_1=r_2=1$$

The general solution of the homogeneous differential equation is

$$y_h=c_1e^x+c_2xe^x$$

Find the particular solution of the non homogeneous differential equation

$$y_p=(Ax^2+Bx+C)e^{5x}$$

$$y_p'=5(Ax^2+Bx+C)e^{5x}$$

$$+(2Ax+B)e^{5x}$$

$$y_p''=25(Ax^2+Bx+C)e^{5x}$$

$$+10(2Ax+B)e^{5x}+2Ae^{5x}$$

Substitute

$$25(Ax^2+Bx+C)e^{5x}$$

$$+10(2Ax+B)e^{5x}+2Ae^{5x}$$

$$-10(Ax^2+Bx+C)e^{5x}-2(2Ax+B)e^{5x}$$

$$+(Ax^2+Bx+C)e^{5x}=x^2e^{5x}$$

$$16Ax^2+(16B+16A)x+16C+8B+2A=x^2$$

$$A=1/16$$

$$B=-1/16$$

$$C=3/128$$

$$y_p=(\dfrac{1}{16}x^2-\dfrac{1}{16}x+\dfrac{3}{128})e^{5x}$$

The general solution of the non homogeneous differential equation is

$$y=c_1e^x+c_2xe^x+(\dfrac{1}{16}x^2-\dfrac{1}{16}x+\dfrac{3}{128})e^{5x}$$