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A question about differential equations

  1. Apr 24, 2012 #1
    Hi,I'm new to these and thus my question might sound stupid: Do differential equations ALWAYS have just one or zero general solutions? I know each diff.equation can have multiple particular solutions, but can it only have one or zero general solutions?
     
  2. jcsd
  3. Apr 24, 2012 #2

    hunt_mat

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    Depending on the DE (if it is linear), you can 2 solutions, add them together and obtain another solution. I think your question relates to either initial conditions or boundary conditions which will lead to uniqueness.
     
  4. Apr 24, 2012 #3

    HallsofIvy

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    That depends upon what you mean by "general solution".

    An example used in many texts is [itex]y'= y^{1/2}[/itex]. That's easily separable so we get [itex]y^{-1/2}dy= dx[/itex] and, integrating, [itex]2y^{1/2}= x+ C[/itex] or [itex]y= (x+ C)^2/4[/itex]. However, it is clear that y(x)= 0, for all x, also satisfies that differential equation. That means that, for y(1)= 0, for example, we can [itex]y= (1+ C)^2/4= 0[/itex] so that C= -1. So that both [itex]y= (x- 1)^2/4[/itex] and y= 0 for all x satisfy both the differential equation and the initial condition.

    I think you need to look at the concepts of "existence and uniqueness" for initial value problems which is probably given in your textbook.
     
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