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TuringTest

  • 4 years ago

y"+y'=3x^2 Can someone explain why, if we use undetermined coefficients on this, we need to guess that Yp has an x^3 in it ? I can't seem to find the rule about this.

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  1. Mertsj
    • 4 years ago
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    Isn't it only sensible that if a sum of two derivatives has an x^2 that the integral would have an x^3? Or am I being too simplistic?

  2. Mertsj
    • 4 years ago
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    Mr. Math should know.

  3. phi
    • 4 years ago
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    the problem is the homogenous solution contains a e^0 i.e. a constant, which matches one of the derivative of the right-hand side

  4. Mr.Math
    • 4 years ago
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    Assume you have some polynomial P(x) as as solution of the given differential equation. Now, the given DE indicates that the sum of the second derivative added to the first derivative gives us a second degree polynomial. That means our assumption should be a polynomial at which its first and second derivative contains a term of x raised to power 2. Does that make sense?

  5. Mr.Math
    • 4 years ago
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    In other words, adding the term with x^3 so that we will get x^2 for y'.

  6. phi
    • 4 years ago
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    if you had e.g. y'' + y' +y= 3x^2 then the particular solution would be order 2

  7. Mr.Math
    • 4 years ago
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    Yes, if we had y in there. But since y' has the largest degree, then it should be of a degree 2. (y' should have a degree 2).

  8. lalaly
    • 4 years ago
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    i explained on the other post Turingtest, please go check it when you see this

  9. amistre64
    • 4 years ago
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    im curious that since the derivative of a function IS a function; why we shouldnt be able to play with this as a: y' + y = 3x^2

  10. amistre64
    • 4 years ago
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    is that possible? or am i being too niave :)

  11. Mr.Math
    • 4 years ago
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    You can deal with it as \(y'+y=x^3\).

  12. Mr.Math
    • 4 years ago
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    I just integrated both sides, as you can see.

  13. TuringTest
    • 4 years ago
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    thanks guys, makes sense :D (I was afk)

  14. phi
    • 4 years ago
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    @ami wolfram says we can get to the same answer, by integrating the solution to your problem http://www.wolframalpha.com/input/?i=y%27%2By%3D+3x%5E2 http://www.wolframalpha.com/input/?i=y%27%27%2By%27%3D+3x%5E2 nice insight!

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