Math Question (Differential Equations)

[eLiu]

Hi all,
this problem really has me stumped...could anyone help me figure out how to start this problem?

This problem deals with Linear, second order homogenous ODEs where b^2 - 4ac <0; that is, the roots of the auxillary equation are complex. Normally, such an equation has the general solution y = exp(Ax)(cos(Bx)+sin(Bx)). However, it is also possible to use the form: F*exp((A+iB)x) + G*exp((A-iB)x).

Show that, in general, F and G must be complex conjugates in order for the solution to be real.

Thanks in advance,
-Eric

 

[HokieESM]

Use Euler's Equation on the two exponentials:

F*e^((A+iB)x)=F*e^(Ax)*(cos(Bx)+i*sin(Bx))
G*e^((A-iB)x)=G*e^(Ax)*(cos(Bx)-i*sin(Bx))

Add the two up and distribute, collecting the Imaginary and Real parts (i'm not going to do ALL the work for you). Its easy to show that F and G must be complex conjugates (i.e. F=a+bi and G=a-bi).