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## Differential equations

### Course: Differential equations>Unit 2

Lesson 2: Complex and repeated roots of characteristic equation

# Complex roots of the characteristic equations 2

What happens when the characteristic equation has complex roots? Created by Sal Khan.

## Want to join the conversation?

• What video, or in what part of the website is the deduction of Euler's formula?
Thank you • I thought C1*i - C2*i = (C1-C2)*i = C*i where C is some arbitrary constant. How did Sal take the i away? •  I assume you are referring to where C1*i - C2*i is turned into C4.
C4 is just a constant that could have any value, even imaginary or complex. Therefore there could be an i hidden inside C4, so the i is not lost, it's just not shown.
• How does this work when you have something greater than a quadratic (order>2)? Because you won't be able to use the quadratic formula and find the Lambda and Mu? • why can you just drop the relationship of c3 being the sum of 2 arbitrary #s and c4, their difference? doesnt c4 have to fall within the range -c3<=c4<=c3? I mean the absolute value of c4 is less than or equal to the absolute value of c3.actually the abs value of c4 could be anything, if the two constants are conjugates. but still, why can you drop the realationship?nevermind,I just realized that i answered my own question, if they are conjugates, then c3 is an abitrary real number, depending on choice of rel part, and c4 is also a arbitray real #, because a pure imaginary # times i is a real #. • From a linear algebra standpoint, if you set it up as a matrix equation:

Ax = b
[ 1 1 ][ c1 ] = [ c3 ]
[ i -i ][ c2 ] = [ c4 ]

A is nonsingular.
(Multiply bottom row2 by i; replace row2 with row2+row1; multiply row2 by 1/2; replace row1 with row1-row2).

Therefore A is also invertible. That means that c1 and c2 can be solved for in terms of c3 and c4, and vice versa. Since c1 and c2 are independent of eac hother, this would imply that there are no dependencies between the variables c3 and c4... I think.
• what if I have an equation with a complex root but not its conjugate root?
(1 vote) • Is any chance, finally after 5 years from first comment about lost imaginary part from particular solution, that you correct this video tutorial and next too.
Someone can say its only particular solution and not full solution of the problem,
or the simply way to avoid complex numbers.
I think its important to show people who interested in second order differential equations that particular solution goes to complex numbers and then one can better
imagine complexity of possible implications. • at around he forget a paranthesis to multiply the e-term with the rest • "The standard pronunciation in English has always been something like oy-ler. We don't have a vowel that corresponds to eu in German, but the next best thing to it is oi. I've never known a mathematician to pronounce Euler as yoo-ler."

I have now! :)   