# Auxiliary Equation with Imaginary Roots

by cronxeh
Tags: auxiliary, equation, imaginary, roots
 PF Patron P: 1,235 I was curious about what class would cover those types of Linear DE w Constant Coeff, particularly Hyperbolic Functions and exp z type of things. I remember my lecturer said back in Intro DE that we only covered first 2 types of Auxiliary Equations - real distinct roots and real repeated ones, but not the imaginary roots because they are 'out of the scope of this course'
 HW Helper P: 2,274 On any Differential Equations course, or ODE course.
 HW Helper Sci Advisor P: 1,123 It's almost exactly the same, but some times you use the different form by the identity: $$e^{x + iy} \equiv e^x \left( \sin y + i \cos y \right)$$
HW Helper
P: 1,593

## Auxiliary Equation with Imaginary Roots

Cronxeh, when you have imaginary roots to an equation, then the solution is of the form:

$$y(x)=c_1e^{(a+bi)x}+c_2e^{(a-bi)x}$$

(and other more complex expressions for repeated complex roots)

You can convert this using Euler's equation:

$$e^{(a+bi)x}=e^{ax}\left(Cos(bx)+iSin(bx)\right)$$

to an expression containing exp's, sin's and cos's. Still have the i though. Can you separate the converted expression into a real part and imaginary part like:

$$y(x)=r(x)+iv(x)$$

If you do, you'll get something like:

$$i(c_1-c_2)$$

as a coefficient on the imaginary part. But that's a constant, call it $k_2$. Now the solution is:

$$y(x)=k_1r(x)+k_2v(x)$$

See how that works?
 PF Patron P: 1,235 Ah thanks. I didnt have time before but now that I'm home I did some digging and found those functions covered in this course: http://www.wellesley.edu/Math/Math20...work/hwk6.html I'm taking Complex Variables in Fall, guess we'll be covering those then
 PF Patron Sci Advisor Thanks Emeritus P: 38,404 " I remember my lecturer said back in Intro DE that we only covered first 2 types of Auxiliary Equations - real distinct roots and real repeated ones, but not the imaginary roots because they are 'out of the scope of this course' " That's a pretty weak D.E. course- even for "Intro". I would hope that your school also has a higher level D.E. course.
 PF Patron P: 1,235 we cover imaginary roots but not from cauchy-euler equations, and this course is only 2 credits and lasts half a semester anyway
HW Helper