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Principle of superposition for a particular case of liner differential equation.

  1. Sep 7, 2011 #1
    Suppose a given second order linear differential equation has two different solutions; then it follows that their linear combination is also a solution. If [tex]y_1(x), y_2(x)[/tex] are two solutions then [tex]y(x)=a_1y_1(x)+a_2y_2(x)[/tex] is also a solution where [tex]a_1, a_2[/tex] are two constants.

    If we consider the equation: [tex]\frac{d^2y(x)}{dx^2}+4\frac{dy(x)}{dx}+4y(x)=0 [/tex], this has two equal solutions, [tex]y_{1}(x)=e^{-2x}=y_2(x)[/tex] but the general solution is given by the nonlinear combination: [tex]y(x)=(a_1+a_2xe^{-2x})[/tex]. This is not in accordance with the principle of superposition for linear equations. Can anybody please explain why this nonlinear combinations is being allowed in this case?

    (Another issue, I have currently switched to ubuntu linux and using firefox. Why am I not able to see the equations properly? All I can see is the latex input commands without the desired output. What has gone wrong? This is happening with other posts containing math formula written in latex format in this forum.)
     
  2. jcsd
  3. Sep 13, 2011 #2

    LCKurtz

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    Yes, for homogeneous equations.
    xe-2x is a solution, which is independent of your y1 and y2 (which aren't themselves independent).
    Yes it is. You have two solutions y1 = y2 and xe-2x and you have a linear combination of them being a solution. The fact that xe-2x itself is a nonlinear function is irrelevant.
     
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