Question

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)

Answer 1

Explanation:

For each of the yp(x) we can deduce the characteristic polynomial of the differential equation

A.

`yp(x)=Ax^2+Bx+C\\m^2=0\\y''=0`

B.

`yp(x)=Ae^(2x)\\(m-2)m=0\\m^2-2m=0\\y''-2y=0`

C.

`yp(x)=Acos2x+Bsin2x\\m_1=2i\\m_2=-2i\\y''+4=0`

D.

`yp(x)=(Ax+B)cos2x+(Cx+D)sin2x\\yp(x)=xcos4x`

hope this helps!!