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'
It's almost exactly the same, but some times you use the different form by the identity: [tex]e^{x + iy} \equiv e^x \left( \sin y + i \cos y \right)[/tex]
Cronxeh, when you have imaginary roots to an equation, then the solution is of the form: [tex]y(x)=c_1e^{(a+bi)x}+c_2e^{(a-bi)x}[/tex] (and other more complex expressions for repeated complex roots) You can convert this using Euler's equation: [tex]e^{(a+bi)x}=e^{ax}\left(Cos(bx)+iSin(bx)\right)[/tex] 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: [tex]y(x)=r(x)+iv(x)[/tex] If you do, you'll get something like: [tex]i(c_1-c_2)[/tex] as a coefficient on the imaginary part. But that's a constant, call it [itex]k_2[/itex]. Now the solution is: [tex]y(x)=k_1r(x)+k_2v(x)[/tex] See how that works?
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/Math208_310sontag/Homework/hwk6.html I'm taking Complex Variables in Fall, guess we'll be covering those then
" 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.
we cover imaginary roots but not from cauchy-euler equations, and this course is only 2 credits and lasts half a semester anyway
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.