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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 3

Lets do an example with initial conditions! Created by Sal Khan.

## Want to join the conversation?

• I enjoy Sal's videos on differential equations, they are as fun as solving puzzles, but I have a question: Are these instructions for solving differential equations are helpful at all aside from solving an equation for the sake of itself? If a computer can solve these equations, wouldn't it be better for me to just learn how to use differential equations for constructing models?
• There are myriad reasons to learn how to solve differential equations by hand.

First, differential equations appear all over science and engineering. If you want to be able to understand an argument or explanation in, say, mechanics (falling objects, oscillations, and pretty much everything else) quantum mechanics (most of these courses start with the Schrodinger wave equation, which involves solving partial differential equations and is prerequisite for any further understanding of the subject) or fluid dynamics and rheology, you're going to have a very hard time picking up even the basics if you have to turn to a computer to do your work for you, and you'll certainly have lots of trouble communicating yourself and understanding what problems are asking if you don't have grounding in this subject, which is assumed in most physics/engineering/biological/chemical curricula beyond the most elementary levels.

Secondly, if you don't have a basic knowledge of how these equations work, so many subjects will just be a black box. It would be like not knowing how to add or multiply because you can just type those things into a calculator. You'll be at a major disadvantage compared to your peers who have learned the subject, and relationships they pick out easily will remain an elusive mystery to you. Knowledge is certainly power. And skill is also power.

Of course, most differential equations actually encountered in the course of doing actual science will require the use of a computer. But that is another matter. Proper and efficient modelling of particular physical or biological systems is something you learn as you come to it. And those skills will assume a standard knowledge of vocabulary, concepts, and the ability to apply them, without which you will always be at a loss for understanding.
• i cant understand...why there is no complex number in solution??/plz help me...which number cancel out "i"??
• i is just a constant, and so it was simply absorbed into c1 and c2.
• Can it happen that I am given two sets of initial conditions, such that they make no sense?, that they can't both be true?, I tried making them but so far I have not been succesful
• do you mean a set of initial conditions that are mutually exclusive of each other? or just a set of initial conditions that cant be true.
• at we found two roots. r= -2+i and r=-2-i why we choose u=1 instead of u=-1?
• It doesn't make a difference, since both constants are arbitrary.
C1 cos -x = C1 cos x, and
C2 sin -x = -C2 sin x. You can just use C2 instead of -C2, since it's arbitrary.
• how will i solve this
D^4 - 5D^2 + 12D + 28=0
find the complex root of this
• You need to do some trick here to solve it, you need to bring in cubes.
D^4 - 5D^2 + 12D + 28=0
=>D^4 +2D^3 - 2D^3 - 4D^2 -D^2 -2D + 14D + 28 = 0
=>D^3(D+2) - 2*D^2(D+2) - D(D+2) + 14(D+2) = 0
=>(D+2)(D^3 - 2*D^2 -D +14) = 0

Now, we can find one value of D which is -2; and need to solve the cubic equation now.
(D^3 - 2*D^2 -D +14) =0
D^3 +2*D^2 -4*D^2 -8D +7D +14 = 0
=> (D+2)(D^2-4D+7)=0 [Skipping some intermediate steps]
So we get, D=-2 (Repeated, D^3 term was '0' in the original equation)
D^2-4D+7=0
Solving this, we will get complex roots since (b^2-4ac)<0,
D = -2, 2+i*sqrt(3), 2-i*sqrt(3) are three roots.

One question may come like why there are three roots but the power of the original expression is 4? Obviously, because there is no CUBIC term in the original expression. We can also say the Algebraic Multiplicity of (-2) is TWO since it is repeated two times.
• Is it possible to have a quadratic equation that has a max value with no real roots
• Do we need to solve for C2 by using the initial condition of the first derivative in this instance only b/c sin(0) becomes 0 and cancels C2, or does this apply for all?
• What if we had a nonhomogeneous differential equation and got complex roots?
(1 vote)
• In place of c2 wont it be ic2?
(1 vote)
• Both are same since it's an ARBITRARY Constant.
(1 vote)
• can you plz suggest me where i could get the videos for geometry of complex no.s ??
(1 vote)
• Have you looked in Sal's complex numbers sections/videos? you will likely find them there:) I always had trouble with complex numbers until a teacher (recently) reminded me that sqrt(-4) is just sqrt(4) * -1, where the -1=i. I haven't had any trouble since.
(1 vote)