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## Differential equations

### Course: Differential equations>Unit 2

Lesson 1: Linear homogeneous equations

# 2nd order linear homogeneous differential equations 1

Introduction to 2nd order, linear, homogeneous differential equations with constant coefficients. Created by Sal Khan.

## Want to join the conversation?

• what is the importance of damping factor in 2nd order system •   I'm an engineering student and I've recently done work on modeling suspension systems for mountain bikes. There is dampening in a lot of systems like that since you don't want your spring to keep osculating after it took an initial blow. So you build in a dampening mechanism.

This is where the By' comes in, since the dampening is often viscous and thereby determined by speed y'.
• The fact that the sum of two solutions to a higher order differential equation is also a solution, is this termed the "superposition principle"? •  Yes, that the sum of arbitrary constant multiples of solutions to a linear homogeneous differential equation is also a solution is called the superposition principle. But if the right hand side of the equation is non-zero, the equation is no longer homogeneous and the superposition principle no longer holds.
• Is the connection between the usage of "homogeneous" as a descriptor for these (linear) differential equations that are set equal to zero and the other variety (where a substitution of f(y/x) is used to render the problem solvable) perhaps not a logical or mathematical one, but a linguistic one? In the former case, we can combine solutions, in the latter the variables are mixed in the solving. Both are varieties of homogenization, although not in the sense a chemist would use the term, no?
(1 vote) • With all due respect, Sal is making a common error of confusing two very similar words. "Homogenous" means "uniform shape" and so far as I can tell the word has no role in differential equations. On the other hand, "homogenEous" (with the extra "e" and five syllables) means "same form" and is relevant to both (although in different ways). The earlier example was of an equation that wasn't separable in x and y but had the same form as a separable equation in v and x when you made the substitution v = y/x. In the case we're studying now, the equation has the same form as a polynomial if we think of the n'th derivative of f like the n'th power of some variable x and then finding the roots of that polynomial to solve the ODE.
• Are there any videos about Partial differential equations? I hear that they are extremely useful in understanding the 'wave equation' thought in Physics better.
Also, out of curiosity, how many solutions can a second-order differential equation have. • Here is a Youtube channel with good PDE videos that I have found helpful in my own studies. https://www.youtube.com/user/commutant . He has several videos on the wave equation that could be beneficial.
Your question is one that mathematicians have struggled with. To prove the existence and uniqueness of solutions to differential equations is still being studied. Only specific kinds of differential equations can be shown to have single solutions, namely, linear, constant coefficient, homogenous equations. Such a proof exists for first order equations and second order equations. I'm not sure what happens higher up. You will have to conduct your own research.
Thumbs up!
• How do you solve equations of inequalities • At Sal concluded that if g(x) and h(x) are both solutions, adding them together also is a solution. Please let me know if I'm understanding this correctly. So when he says one of those functions "is a solution", in this case, since he's speaking of homogenous equations, he's basically saying "is equal to 0"? So, then, when he's saying "g(x) + h(x) is a solution" he is pretty much saying "0+0=0"? Am I correct in my understanding? Thanks! • At wat dose he mean 2nd ordeal diferntial equations
(1 vote) • The order of a differential equation is the highest-order derivative that it involves. Thus, a second order differential equation is one in which there is a second derivative but not a third or higher derivative.

Incidentally, unless it has been a long time since you updated your profile, you might be in over your head on this one. I might recommend taking a while to learn differential and integral calculus before you try to tackle differential equations.
• 5/9(10^1+10^2+10^3+.....+10^n-n) = sum ?
How can i get the summation of this series using c program?
please share me this programe..... its urgent
thank u
(1 vote) • Will second order homogeneous differential equations always have solutions in terms of e^x?
(1 vote) • No. Only constant coefficient second order homogeneous differential equations where the associated auxiliary equation has two distinct real roots will have both solutions being e^mx, where m is a real number.

Since there are other types of second order homogeneous DEs, like Cauchy Euler as an example, where the solutions are not e^mx, you won't always get e^mx solutions. 