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Question about linear ODEs

  1. Feb 26, 2013 #1
    So, this is probably really easy, but it's been bugging me...is the following differential equation linear?

    [tex]
    e^{y'' + y} = 12
    [/tex]

    'Cause can't you just take logarithms on both sides and get it to be

    [tex]
    y'' + y = \log 12
    [/tex]

    I guess the question I'm trying to ask is...what operations are you allowed to take an ODE through in trying to put it in linear form?
     
  2. jcsd
  3. Feb 26, 2013 #2
    Well, the first equation is not a linear ODE, but it can be linearized easily. Most strategies for solving ODE's are based around a transformation to a form that is easily solvable. For first order equations, you usually try to transform the ODE to an exact ODE by finding an integrating factor.

    Another example, this equation:

    dy/dx = 1/(x-y(x))

    is inverse-linear. You can linearize it if you change the dependent and independent variables x->x(y) and y(x)->y and you will get:

    dy/dx = 1/(x(y) - y)
    dx/dy = x(y) - y, or:
    x' = x - y
     
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