# Auxiliary Equation with Imaginary Roots

1. ### cronxeh

1,202
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'

2. ### Pyrrhus

2,205
On any Differential Equations course, or ODE course.

3. ### Zurtex

1,123
It's almost exactly the same, but some times you use the different form by the identity:

$$e^{x + iy} \equiv e^x \left( \sin y + i \cos y \right)$$

4. ### saltydog

1,581
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?

1,202
6. ### HallsofIvy

41,061
Staff Emeritus
" 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.

7. ### cronxeh

1,202
we cover imaginary roots but not from cauchy-euler equations, and this course is only 2 credits and lasts half a semester anyway

8. ### Zurtex

1,123
We covered exactly the same in Calc A at University. Excpet is was all done in 30 miniuites and our Tutor is so slow at ocvering stuff it missed out loads. I'm so glad I did Further Maths at A Level.