Need help understanding linear equations of higher order

[bitrex]

I'm having some trouble getting my head around the concept of multiple solutions of differential equations of higher order, that is the general solution to a linear homogeneous equation is a linear combination of constants and solutions like y(1)C1 + y(2)C2 +y(n)C(n) where N is the order of the differential equation. I understand that there will be multiple constants because even if it's in a roundabout way to solve the equation n integrations are neccessary, but for say a second order equation why will there be 2 solutions, and not one? Or three?

 

[tiny-tim]

… for say a second order equation why will there be 2 solutions, and not one? Or three?

Hi bitrex!

Because an equation (∑ a_n(d/dx)ⁿ)y = 0

can be factored as (d/dx - c₁)(d/dx - c₂) … (d/dx - c_n)y = 0,

which obviously is the same as the n individual equations (d/dx - c₁)y = 0, (d/dx - c₂)y = 0, … (d/dx - c_n)y = 0

 

[HallsofIvy]

One of the things that should be shown early in an Introductory Differential Equations course is this:

"The set of all solutions to a homogeneous linear differential equation forms a vector space of dimension n".

Of course, saying that a vector space has dimension n means that there exist a set of n vectors, $\left{v_1, v_2, ..., v_n\right}$, a basis for the vector space, such that any vector, v, in the vector space can be written as a unique linear combinations of the basis vectors: $v= a_1v_1+ a_2v_2+ ...+ a_nv_n$ for numbers $a_1, a_2, ..., a_n$. The $y(1), y(2), ..., y(n)$ functions you list are the "basis vectors" for this vector space.

 

[g_edgar]

Because an equation (∑ a_n(d/dx)ⁿ)y = 0

can be factored as (d/dx - c₁)(d/dx - c₂) … (d/dx - c_n)y = 0,

He didn't say his DE has constant coefficients

 

[bitrex]

Thank you all for your replies. I think the way that g_edgar puts it makes the most sense, in that I guess one can think of the multiple solutions as roots of the polynomial operator working on the differential of y. I think Halls' explanation would be the most elegant, if I were capable of understanding it I don't have much experience with linear algebra, is there a way you could elaborate by way of example? I checked out the Wikipedia entry on basis vectors and it's unfortunately it seems to be one of those Wikipedia mathematics articles that's useless unless you already have familiarity with the subject. Apparently there's some sort of war going on between Wikipedia editors who want to make the Wikipedia mathematics entries more of a teaching resource, and those who want to keep them strictly encyclopedic. If I really wanted to feel intellectually inadequate, I'd just go to Wolfram Mathworld... In any case, there appears to be a link to a lecture at MIT fairly early on in the Linear Algebra series that covers basis vectors so I'll have a look at that and then see if the vector space explanation makes more sense.

 

[srijithju]

I think you can also probably find a link to An MIT lecture which explains why an nth order homogeneous diff equation has n independent solutions... there is a whole set of video lectures on differentail equations