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Homework Help: Are these two general solutions or one general/two particular solutions?

  1. Apr 22, 2012 #1

    s3a

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    1. The problem statement, all variables and given/known data
    The problem and its answers are attached.

    I prefer the first one as (same thing but isolated for y):

    y_1(x) = [k_1 * x + x * ln(x)]/[k2 - ln(x)]


    2. Relevant equations
    First order homogeneous differential equation.
    y = xv, dy/dx = v + x dv/dx


    3. The attempt at a solution
    I solved this successfully. Given that at some point in the solution of the problem for solving y_1(x), I get a denominator with (x+y), it is my "mathematical duty" to check if x + y = 0 => y = -x is a solution to the differential equation and it is. Because of the nature in which I got this y = -x solution, I am confused as to whether I treat this as one of the two alternatives for a general solution or both together span the one and only general solution.

    Any input would be greatly appreciated!
    Thanks in advance!
     

    Attached Files:

  2. jcsd
  3. Apr 22, 2012 #2

    Dick

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    Science Advisor
    Homework Helper

    Your equation isn't linear, is it? It has a y^2 in it. That would mean a linear combination of solutions isn't necessarily a solution.
     
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