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I've been reading intros to ODE and the problem of terminology has overwhelmed me.

As far as I understand:

  1. n-Parameter family of solutions to ODE is a solution in a form $c_1y_1+c_2y_2+...+c_ny_n$
  2. singular solution to ODE is a solution which can't be obtained from (1) (no combination of $\{c_1;c_2;...;c_n\}$ can give you this singular solution).

  3. general solution is: a) an n-Parameter family solutions + all singular solutions OR b) an n-Parameter family solutions only when there are no singular solutions exist. Which one is right?

  4. fundamental solution...?

  5. fundamental set of solutions as far as I know is a set formed by taking solutions from (1) $\{y_1;y_2;...;y_n\}$

What's the point in talking about "fundamental solution" when we have a general one? if I'm right about (5) Why we don't call it "general set of solutions"? Why invent a new phrase - fundamental set?

coobit
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2 Answers2

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"I've been reading intros to ODE and the problem of terminology has overwhelmed me.

As far as I understand: 1.n-Parameter family of solutions to ODE is a solution in a form cy1+ky2" No. That is a "two parameter" solution to a (linear) second order equation. An "n-parameter family" to a (linear) nth order equation would be a linear combination of n "fundamental solutions".

"2.singular solution to ODE is a solution which can't be obtained from (1)"

Yes, but such singular solutions appear only in non-linear equations.'

"3.general solution is a n-Parameter family of solutions when singular solutions are absent." Yes, that follows immediately from the definition of "singular solution". If there are no singular solutions, that is, no solutions that are not of that form, then all solutions are of that form!

"4.fundamental solution...?" A "fundamental solution" to a linear, homogeneous, nth order differential equation (at x= a) is a solution to the equation such that y(a)= 1, y'(a)= 0, ..., $y^{(n-1)}(a)= 0$ or such that y(a)= 0, y'(a)= 1, y''(a)= 0 ..., $y^{(n- 1)}(a)= 0$, etc.

"5.fundamental set of solutions...?" A set of fundamental solutions all the way from "y(a)= 1, y'(a)= 0, ..., $y^{(n-1)}= 0$" to "y(a)= 0, y'(a)= 0, ..., $y^{(n-1)}= 1".

"What's the point in talking about "fundamental solution" when we have a general one?" If we already have a general solution then there would be no point at all. But before we have a "general solution" we must know such a thing [b]exists[/b]. Have you read through the basic "existence and uniqueness theorem" for first order equations? For nth order equations? Those show that the "fundamental solutons" [b]exist[/b] for each of the "initial value problems" above and the fundamental solutions can then be used to construct the "general solution". Do you know what "linear" and "homogeneous" mean here? Do you understand that the "general solution", as given, only exists for linear homogeneous equations? If not, you are going through the book to quickly and should go back and start again.

user247327
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  • Mmm.... Your definition of fundamental set and solutions is not what I'm used to. My books say that if we have general solution, e.g. cy1+ky2, then the "fundamental set" is a collection of {y1;y2;...}. – coobit May 15 '16 at 16:31
  • That doesn't make sense. The "general solution" to any, say, second order equation can be written as a sum of two functions in an infinite number of ways so it would not make sense to talk about "the" fundamental set in that sense. For example, the "general solution" to the differential d^2y/dt^2- y= 0 can be written either [tex]Ae^x+ Be^{-x}[/tex] or [tex]C cosh(x)+ D sinh(x)[/tex]- all functions of one form can be written in the other form. But the sets [tex]{e^x, e^{-x}}[/tex] and [tex]{cosh(x), sinh(x)}[/tex] are different sets of functions. – user247327 May 16 '16 at 16:21
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After countless articles I think I've managed to unearth the truth.

General Solution:

The word general here means that a solution to ODE has some parameters in it. So it is "general" in that sence but not a full solution to a given ODE, since there could be some singular solutions outside of the general form. So the terminology here is most unfortunate!

Fundamental solution:

A General solution to an ODE from which we can obtain every other solution is called a fundamental.

coobit
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