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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 1

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

## Want to join the conversation?

• Do complex roots have any application to economics/finances? I's love to watch all the videos, they're awesome, but, due to lack of time, I decided to stick to the ones in my area.
• The complex components in the solution to differential equations produce fixed regular cycles. Arbitrage reactions in economics and finance imply that these cycles cannot persist, so this kind of equation and its solution are not really relevant in economics and finance. Think of the equation as part of a larger system, and think of the equation's coefficients (perhaps order too) depending on the results from the other equations.
• the last part...shouldn't the ux be exponent? instead of put it after sin and cos
• If you remember the formula e^ix = cos(x) + i sin(x), you'll see that the arguments of sine and cosine are the same as what e is raised to (excluding the x). So if e was raised to i and some arbitrary constant or variable, let's say we use pi (so I'm talking about e^i*pi), then the arguments of sine and cosine would be pi (i.e. cos(pi) + i sin(pi) ). This interesting result also yields the famous equation e^(i*pi) = -1
• Isn't Euler pronounced like "Oiler" instead of "Yooler"? At least that's how I've been taught through school.
• i have a question that if the equation is y"+y+3y=0, why is the characteristic equation is r^2+r+3=0 rather than r^2+4=0??
• The characteristic equation of `yʺ + yʹ + 3y = 0` is `r² + r + 3 = 0`.

The characteristic equation of `yʺ + y + 3y = 0` is indeed `r² + 4 = 0`.
• where can i learn about imaginary numbers?
• At , I understand that e^i*whatever is the cos(whatever) + i sin(whatever), but would it be wrong to say that since e^i*x = cos(x)+isin(x), e^i*x*u = (cos(x)+isin(x))^u?
• how do you use your calculator to find roots of an equation?
• If you are talking about roots for quadratic equations, you can just plug in the required numbers into the quadratic equation. If you are talking about n-order equations, you can either factor them out to give you the roots or graph them to show you the roots. Some of the equations do have complex roots, though, so you will have to watch out for those. Hope this helped.