Match the following guess solutions yp for the method of und

Match the following guess solutions yp for the method of undetermined coefficients with the second-order nonhomogeneous linear equations below.

A. yp(x)=Ax2+Bx+C, B. yp(x)=Ae2x, C.yp(x)=Acos2x+Bsin2x,
D. yp(x)=(Ax+B)cos2x+(Cx+D)sin2x E. yp(x)=Axe2x, and F.yp(x)=e3x(Acos2x+Bsin2x)

1. d2ydx2+4y=xx220

2. d2ydx2+6dydx+8y=e2x

3. y+4y+20y=3sin2x

4. y2y15y=3xcos2x

1 ) Given that

(d²y/dx²) + 4y = x - x² + 20

(d²y/dx²) + 4y = - x² + x + 20

For a non homogeneous part - x² + x + 20 , we assume the perticular solution is

y_p(x) = Ax² + Bx + C

2 ) Given that

d²y/dx² + 6dy/dx + 8y = e^2x

For a non homogeneous part e^2x , we assume the perticular solution is

y_p(x) = Ae^2x

3 ) Given that

y + 4y + 20y = 3sin(2x)

For a non homogeneous part 3sin(2x) , we assume the perticular solution is

y_p(x) = Acos(2x)+Bsin(2x)

4 ) Given that

y 2y 15y = 3xcos(2x)

For a non homogeneous part 3xcos(2x) , we assume the perticular solution is

y_p(x) = (Ax+B)cos2x+(Cx+D)sin2x