Question

Find the general solution of the following differential equations using method of undetermined coefficients

(i) y''−7y'+10y =20e^5x −10

(ii) y''−2y'= x^2 +5x−2

(iii) y''+ 9y'+14y = 20+e^2x +e^−2x

(iv) y''−2y'+y =e^x+x^2

(v) y''−y'=cosx+5.

Answer 1

(i)

Corresponding homogeneous differential equation

$$y''−7y'+10y =0$$

Characteristic (auxiliary) equation

$$r^2-7r+10=0$$

$$r_1=2,r_2=5$$

The general solution of the homogeneous differential equation is

$$y_h=c_1e^{2x}+c_2e^{5x}$$

Find the particular solution of the non homogeneous differential equation

$$y_p=Axe^{5x}+B$$

$$y_p'=5Axe^{5x}+Ae^{5x}$$

$$y_p''=25Axe^{5x}+10Ae^{5x}$$

Substitute

$$25Axe^{5x}+10Ae^{5x}-35Axe^{5x}-7Ae^{5x}$$

$$+10Axe^{5x}+10B=20e^{5x}-10$$

$$A=20/3, B=-1$$

$$y_p=\dfrac{20}{3}xe^{5x}-1$$

The general solution of the non homogeneous differential equation is

$$y=c_1e^{2x}+c_2e^{5x}+\dfrac{20}{3}xe^{5x}-1$$

(ii)

Corresponding homogeneous differential equation

$$y''−2y' =0$$

Characteristic (auxiliary) equation

$$r^2-2r=0$$

$$r_1=0,r_2=2$$

The general solution of the homogeneous differential equation is

$$y_h=c_1+c_2e^{2x}$$

Find the particular solution of the non homogeneous differential equation

$$y_p=Ax^3+Bx^2+Cx$$

$$y_p'=3Ax^2+2Bx+C$$

$$y_p''=6Ax+2B$$

Substitute

$$6Ax+2B-6Ax^2-4Bx-2C$$

$$=x^2+5x-2$$

$$-6A=1$$

$$6A-4B=5$$

$$2B-2C=-2$$

$$A=-1/6, B=-3/2, C=-1/2$$

$$y_p=-\dfrac{1}{6}x^3-\dfrac{3}{2}x^2-\dfrac{1}{2}x$$

The general solution of the non homogeneous differential equation is

$$y=c_1+c_2e^{2x}-\dfrac{1}{6}x^3-\dfrac{3}{2}x^2-\dfrac{1}{2}x$$

(iii)

Corresponding homogeneous differential equation

$$y''+9y'+14y =0$$

Characteristic (auxiliary) equation

$$r^2+9r+14=0$$

$$r_1=-7,r_2=-2$$

The general solution of the homogeneous differential equation is

$$y_h=c_1e^{-7x}+c_2e^{-2x}$$

Find the particular solution of the non homogeneous differential equation

$$y_p=Ae^{2x}+Bxe^{-2x}+C$$

$$y_p'=2Ae^{2x}-2Bxe^{-2x}+Be^{-2x}$$

$$y_p''=4Ae^{2x}+4Bxe^{-2x}-4Be^{-2x}$$

Substitute

$$4Ae^{2x}+4Bxe^{-2x}-4Be^{-2x}$$

$$+18Ae^{2x}-18Bxe^{-2x}+9Be^{-2x}$$

$$+14Ae^{2x}+14Bxe^{-2x}+14C$$

$$=20+e^{2x} +e^{−2x}$$

$$36A=1$$

$$5B=1$$

$$14C=20$$

$$y_p=\dfrac{1}{36}e^{2x}+\dfrac{1}{5}xe^{-2x}+\dfrac{10}{7}$$

The general solution of the non homogeneous differential equation is

$$y=c_1e^{-7x}+c_2e^{-2x}+\dfrac{1}{36}e^{2x}+\dfrac{1}{5}xe^{-2x}+\dfrac{10}{7}$$

(iv)

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

$$y_p=Ax^2e^{x}+Bx^2+Cx+D$$

$$y_p'=Ax^2e^x+2Axe^x+2Bx+C$$

$$y_p''=Ax^2e^x+4Axe^x+2Ae^x+2B$$

Substitute

$$Ax^2e^x+4Axe^x+2Ae^x+2B$$

$$-2Ax^2e^x-4Axe^x-4Bx-2C$$

$$+Ax^2e^{x}+Bx^2+Cx+D$$

$$=e^x+x^2$$

$$2A=1$$

$$B=1$$

$$C=4$$

$$D=6$$

$$y_p=\dfrac{1}{2}x^2e^{x}-x^2+4x+6$$

The general solution of the non homogeneous differential equation is

$$y=c_1e^x+c_2xe^x+\dfrac{1}{2}x^2e^{x}-x^2+4x+6$$

(v)

Corresponding homogeneous differential equation

$$y''−y' =0$$

Characteristic (auxiliary) equation

$$r^2-r=0$$

$$r_1=0,r_2=1$$

The general solution of the homogeneous differential equation is

$$y_h=c_1+c_2e^{x}$$

Find the particular solution of the non homogeneous differential equation

$$y_p=A\cos x+B\sin x+Cx$$

y''−y'=cosx+5.

$$y_p'=-A\sin x+B\cos x+C$$

$$y_p''=-A\cos x-B\sin x$$

Substitute

$$-A\cos x-B\sin x+A\sin x+B\cos x-C$$

$$=\cos x+5$$

$$A=-1/2, B=-1/2, C=-5$$

$$y_p=-\dfrac{1}{2}\cos x-\dfrac{1}{2}\sin x-5x$$

The general solution of the non homogeneous differential equation is

$$y=c_1+c_2e^{x}-\dfrac{1}{2}\cos x-\dfrac{1}{2}\sin x-5x$$