# A quick question on complimentary functions

• Jun 8th 2009, 02:06 AM
craig
A quick question on complimentary functions
Hi.

If the auxiliary equation of a second order differential equation has 2 complex repeated roots, would I still use:

$\displaystyle e^{\alpha x}(A \cos{\beta x} + B\sin{\beta x})$

Just in the form:

$\displaystyle A \cos{\beta x} + B\sin{\beta x}$?

Thanks
• Jun 8th 2009, 02:16 AM
craig
Doesn't matter, found out that you use $\displaystyle e^{\alpha x}(A \cos{\beta x} + B\sin{\beta x})$
• Jun 8th 2009, 02:53 AM
mr fantastic
Quote:

Originally Posted by craig
Hi.

If the auxiliary equation of a second order differential equation has 2 repeated roots, would I still use:

$\displaystyle e^{\alpha x}(A \cos{\beta x} + B\sin{\beta x})$

Just in the form:

$\displaystyle A \cos{\beta x} + B\sin{\beta x}$?

Thanks

If the root is repeated then the general solution is $\displaystyle y = (Ax + B) e^{\lambda x}$.
• Jun 8th 2009, 02:54 AM
mr fantastic
Quote:

Originally Posted by craig
Doesn't matter, found out that you use $\displaystyle e^{\alpha x}(A \cos{\beta x} + B\sin{\beta x})$

This is the general solution for when the roots are not real, that is, the roots have the form $\displaystyle \lambda = \alpha \pm i \beta$.
• Jun 8th 2009, 05:41 AM
craig
Hi thanks for the reply, there was actually a typo in my original post, it was for roots such as $\displaystyle \pm 2i$.

Have fixed the typo now.
• Jun 9th 2009, 08:38 PM
Sampras
Quote:

Originally Posted by craig
Hi thanks for the reply, there was actually a typo in my original post, it was for roots such as $\displaystyle \pm 2i$.

Have fixed the typo now.

Assume the coefficients are real. Then the roots come in pairs. I would think the general solution would be $\displaystyle y = c_{1}e^{ax} \cos bx+c_{2}xe^{ax} \cos bx+c_{3}e^{ax} \sin bx + c_{4}xe^{ax} \sin bx$ for $\displaystyle \lambda = a \pm bi$ if degree of multiplicity is $\displaystyle 2$.
• Jun 14th 2009, 02:46 AM
HallsofIvy
Quote:

Originally Posted by Sampras
Assume the coefficients are real. Then the roots come in pairs. I would think the general solution would be $\displaystyle y = c_{1}e^{ax} \cos bx+c_{2}xe^{ax} \cos bx+c_{3}e^{ax} \sin bx + c_{4}xe^{ax} \sin bx$ for $\displaystyle \lambda = a \pm bi$ if degree of multiplicity is $\displaystyle 2$.

No, that the same as $\displaystyle (c_1+ c_3)e^{ax}cos bx+ (c_2+ c_4)e^{ax}sin bx$ so you have nothing new. For a+ bi a double root of the characteristic equation, you want $\displaystyle y= e^{ax}(c_1cos(bx)+ c_2sin(bx))+ xe^{ax}(c_3cos(bx)+ c_4sin(bx))$.

That is, multiply the "base" solution by x as mr. fantastic said.