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Linear Ordinary Differential Equation: Definition

  1. May 3, 2015 #1
    1. The problem statement, all variables and given/known data
    The website says this:
    "It is Linear when the variable (and its derivatives) has no exponent or other function put on it.
    So no y2, y3, √y, sin(y), ln(y) etc, just plain y (or whatever the variable is).
    More formally a Linear Differential Equation is in the form:
    dy/dx + P(x)y = Q(x)"

    My question is whether what makes it a linear differential equation the fact that nothing on the other side of the equals sign from Q(x) has any degree higher than one or whether it is a linear differential equation because the differential doesn't have a degree higher than 1; for example, is this a linear differential equation?
    dy/dx + y^3 = Q(x)
    What about this one?
    dy/dx + P(x)y = Q(x)^2
    2. Relevant equations

    3. The attempt at a solution
    http://en.wikipedia.org/wiki/Linear_differential_equation
    http://www.mathsisfun.com/calculus/differential-equations.html

    Note: Sorry on the title. I meant Linear Ordinary Differential Equation. Partial should not be in there.
     
    Last edited: May 3, 2015
  2. jcsd
  3. May 3, 2015 #2

    Mark44

    Staff: Mentor

    Maybe a better way to define a linear differential equation is that it will consist only of a linear combination of the dependent variable and its derivatives. In other words, an equation that consists of either a sum of constant multiples of y, y', y'', etc. or a sum where y, y', y'' etc. are multiplied by function of the independent variable.
    No, not linear, because of the y3 term.
    Yes, linear. It doesn't matter that y is multiplied by P(x) or that we have [Q(x)]2.
    I removed "Partial" from the thread title.
     
  4. May 3, 2015 #3
    Thanks for the help Mark44. I understand now.
     
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