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2nd ODE, Reduction of Order, Basis known

  1. Aug 25, 2010 #1
    I have a homework problem where I am to find y_2 for a 2nd ODE, with y_1=x.

    I'm familiar with the process of:

    let y_2 = ux

    y_2- = u'x u

    y_2'' = 2u' + u''x

    substituting these terms into the 2ODE, then letting u' = v.

    When integrating v and u' to solve for u, do I need to include integration constants at both steps?

    I have a textbook that suggests that integration constants can be made redundant by choosing the 1 and 0 as their values.

    However, I have lecture notes which seem to include integration constants in the final solution.

    What's the correct approach?

  2. jcsd
  3. Aug 25, 2010 #2


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    Science Advisor

    It depends upon what you want. If you include the constants of integration you will get something like: u(x)= Cf(x)+ D with C and D constants. Then, from y= ux, your solution is y(x)= (Cf(x)+ D)x= Cf(x)x+ Dx, the general solution to the original equation.

    If the equation is linear and you do not include the constants of integration, you will get only u(x)= f(x) so that y(x)= f(x)x. But then you can say "I now have two independent solutions, f(x)x and x, to this linear equation so the general solution is Cf(x)x+ Dx" for any constants C and D. That gives exactly the same solution.
  4. Aug 26, 2010 #3
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