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### 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?

• 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.
• 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
• Differential equations describe the way things change. Framing that as an inequality is trying to solve how they don't change. The solution space is unbounded, everything except how they do change is a solution. That may sound like it describes a solution in negative terms, but doesn't seem to lend itself to a process to determine it
• 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!
• Yes, I think you have it exactly right.

You can see that when you plug solutions in the terms can be easily separated which really simplifies things because you can just sum all your solutions. You'll use this same idea later with non-Homogeneous equations.
• At wat dose he mean 2nd ordeal diferntial equations
• 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
• Order is the highest derivative present in the equation. Degree is the exponent of the highest derivative term.

(y'')^3 + y' + 1 = 0 is second order due to y'' and is 3rd degree since the highest derivative is raised to the 3rd power.
• Will second order homogeneous differential equations always have solutions in terms of e^x?
• 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.