Answer to Question #268519 in Differential Equations for pia

Question

a.) 2y′′′−7y′′+12y′+8y=0

b.) y"-2y'+5y=e^xsinx

c.) y′' + y = 2xsinx

Answer 1

a)

$$2y'''-7y''+12y'+8y=0$$

Auxiliary equation (or characteristic equation)

$$2r^3-7r^2+12r+8=0$$

$$2r^2(r+\dfrac{1}{2})-8r(r+\dfrac{1}{2})+16(r+\dfrac{1}{2})=0$$

$$2(r+\dfrac{1}{2})(r^2-4r+8)=0$$

$$2(r+\dfrac{1}{2})((r-2)^2+4)=0$$

$$r_1=-\dfrac{1}{2}, r_2=2-2i, r_3=2+2i$$

The general solution of the homogeneous differential equation is

$$y=c_1e^{-x/2}+e^{2x}(c_2\cos(2x)+c_3\sin(2x))$$

b)

$$y''-2y'+5y=e^x\sin x$$

The corresponding homogeneous differential equation is

$$y''-2y'+5y=0$$

Auxiliary equation (or characteristic equation)

$$r^2-2r+5=0$$

$$(r-1)^2=-4$$

$$r_1=1-2i, r_2=1+2i$$

The general solution of the homogeneous differential equation is

$$y_h=e^{x}(c_1\cos(2x)+c_2\sin(2x))$$

Find the particular solution of the nonhomogeneous differential equation in form

$$y_p=Ae^x\cos x+Be^x\sin x$$

$$y_p'=Ae^x\cos x-Ae^x\sin x+Be^x\sin x+Be^x\cos x$$

$$y_p''=Ae^x\cos x-2Ae^x\sin x-Ae^x\cos x$$

$$+Be^x\sin x+2Be^x\cos x-Be^x\sin x$$

Substitute

$$-2Ae^x\sin x+2Be^x\cos x-2Ae^x\cos x$$

$$+2Ae^x\sin x-2Be^x\sin x-2Be^x\cos x$$

$$+5Ae^x\cos x+5Be^x\sin x=e^x\sin x$$

$$3B=1$$

$$A=0$$

$$y_p=\dfrac{1}{3}e^x\sin x$$

The general solution of the nonhomogeneous differential equation is

$$y=e^{x}(c_1\cos(2x)+c_2\sin(2x))+\dfrac{1}{3}e^x\sin x$$

c)

$$y'' + y = 2x\sin x$$

The corresponding homogeneous differential equation is

$$y''+y=0$$

Auxiliary equation (or characteristic equation)

$$r^2+1=0$$

$$r_1=i, r_2=i$$

The general solution of the homogeneous differential equation is

$$y_h=c_1\cos x+c_2\sin x$$

Find the particular solution of the nonhomogeneous differential equation in form

$$y_p=(Ax^2+Bx+C)\cos x+(Dx^2+Ex+F)\sin x$$

$$y_p'=-(Ax^2+Bx+C)\sin x+(2Ax+B)\cos x$$

$$+(Dx^2+Ex+F)\cos x+(2Dx+E)\sin x$$

$$y_p''=-(Ax^2+Bx+C)\cos x-2(2Ax+B)\sin x$$

$$+2A\cos x-(Dx^2+Ex+F)\sin x$$

$$+2(2Dx+E)\cos x+2D\sin x$$

Substitute

$$-(Ax^2+Bx+C)\cos x-2(2Ax+B)\sin x$$

$$+2A\cos x-(Dx^2+Ex+F)\sin x$$

$$+2(2Dx+E)\cos x+2D\sin x$$

$$+(Ax^2+Bx+C)\cos x+(Dx^2+Ex+F)\sin x$$

$$=2x\sin x$$

$$-2(2Ax+B)\sin x+2A\cos x$$

$$+2(2Dx+E)\cos x+2D\sin x=2x\sin x$$

$$A=-\dfrac{1}{2}$$

$$B=D=0$$

$$E=-A=\dfrac{1}{2}$$

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

The general solution of the nonhomogeneous differential equation is

$$y=c_1\cos x+c_2\sin x-\dfrac{1}{2}x^2\cos x+\dfrac{1}{2}x\sin x$$