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Need help understanding Superposition Principle

  1. Sep 28, 2013 #1
    need help understanding "Superposition Principle"..!!

    hello everyone..
    if we have a function y=f(x) then in-order to prove linearity we try to justify according to superposition principle as :
    let x1 and x2 be two inputs then f(x1+x2)=f(x1)+f(x2)
    please correct me if i am wrong upto here..
    now what if we have more than two variables..let's say we have three variables two independent and one dependent
    now we have function z=g(x,y)..now in-order to prove linearity for function involving more than two variables can i say this that for g(x,y) to be linear g(x1+x2,y1+y2)=g(x1,y1)+g(x2,y2)..??
    and if this isn't the correct way for proving linearity in functions involving more than two variables..then please justify the correct method along with examples.
     
  2. jcsd
  3. Sep 28, 2013 #2

    Office_Shredder

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    If you have a function of multiple variables, you typically want what's called multilinearity - that the function is linear in each variable. For example, g(x1+x2,y) = g(x1,y) + g(x2,y) and g(x,y1+y2) = g(x,y1) + g(x,y2). In this case you should be able to figure out what g(x1+x2,y1+y2) is equal to (it's not what you wrote).

    What your g is satisfying is that it is linear in the single input (x,y), which may be what you're looking for.
     
  4. Sep 28, 2013 #3
    hello..
    i am understanding a little bit now but if i have to say linearity of functions involving more than two variables then i can't always refer to superposition principle or is there any superposition involving more than two variables..!!
    and if i have to consider the linearity among differential equations as in linear differential equation then what would be method to justify this..can this multi-linearity principle also holds for differential equation..?
     
  5. Sep 28, 2013 #4

    Office_Shredder

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    I don't understand what your question is, can you give a specific example?
     
  6. Sep 28, 2013 #5
    i mean we mention differential equation to be linear..as linear differential equation..

    and for the example if we take this LDE dy/dt+(x^2)*y=0
    it is LDE as for the dependent variable and its deriavtive is in first degree and are not multipled together..please let me know i am wrong..!!
    then can we apply the superposition principle on this one to justify its linearity
    for this one if i have y1 for x1 and y2 for x2 then if i input x1+x2 will i get y as y1+y2..?? acc. to superposition principle..can i really justify its linearity with superposition principle of f(x1+x2)=f(x1)+f(x2)...?
     
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