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Math Help - Dif EQ

  1. #1
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    Dif EQ

    Another Dif EQ, yay.

    I have to solve the following Dif EQ, and give the interval for which the largest general solution is defined:

    dP/dt + 2tP = P + 4t - 2

    So I don't think I can do separation of variables, so I tried integrating factors...

    P(x) = 2t

    M(x) = e^(int(2t)dt) = e^(2t^3/3)

    So multiply M(x) to each term:

    e^(2t^3/3)*dP/dt + e^(2t^3/3)*2tP = e^(2t^3/3)*P + e^(2t^3/3)*4t - 2*e^(2t^3/3)

    So this looks very messy now...

    Again, it's the part after this that I get stuck.
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    Upon re-examining this,

    Would it be:

    int((e^(2t^3/3)*P)') = int(e^(2t^3/3)*P + e^(2t^3/3)*4t - 2*e^(2t^3/3))

    => (e^(2t^3/3)*P) = int(e^(2t^3/3)*P + e^(2t^3/3)*4t - 2*e^(2t^3/3))
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Ideasman View Post
    Another Dif EQ, yay.

    I have to solve the following Dif EQ, and give the interval for which the largest general solution is defined:

    dP/dt + 2tP = P + 4t - 2

    So I don't think I can do separation of variables, so I tried integrating factors...

    P(x) = 2t
    you are wrong since this line

    Quote Originally Posted by Ideasman View Post
    Upon re-examining this,

    Would it be:

    int((e^(2t^3/3)*P)') = int(e^(2t^3/3)*P + e^(2t^3/3)*4t - 2*e^(2t^3/3))

    => (e^(2t^3/3)*P) = int(e^(2t^3/3)*P + e^(2t^3/3)*4t - 2*e^(2t^3/3))
    thus, this is wrong as well
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    Quote Originally Posted by Jhevon View Post
    you are wrong since this line
    But why? I thought you took the stuff that wasn't on the dependent variable and let that equal P(x). Is it because of the RHS of the equation? Do I need to move things first? Suppose it was just

    dP/dt + 2tP = 0. Would P(x) be 2t then?
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Ideasman View Post
    But why? I thought you took the stuff that wasn't on the dependent variable and let that equal P(x). Is it because of the RHS of the equation? Do I need to move things first?
    you had P' + 2tP = P + 4t - 2

    \Rightarrow P' + 2tP - P = 4t - 2

    \Rightarrow P' + (2t - 1)P = 4t - 2

    now what do you think P(x) is?

    (in this case, using "P" gets confusing, because that's the variable we're working with in the first place, but oh well)

    Suppose it was just

    dP/dt + 2tP = 0. Would P(x) be 2t then?
    yes, but you would not use the integrating factor in this case, this is a homogeneous first order DE, it is separable
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    Quote Originally Posted by Jhevon View Post
    you had P' + 2tP = P + 4t - 2

    \Rightarrow P' + 2tP - P = 4t - 2

    \Rightarrow P' + (2t - 1)P = 4t - 2

    now what do you think P(x) is?
    2t - 1

    Okay, so if P(x) = 2t - 1

    That means M(x) = e^(int(2t - 1)dt) = e^(t^2 - t)

    Multiply M(x) by each term:

    e^(t^2 - t)*P' + e^(t^2 - t)*P = e^(t^2 - t)*4t - 2*e^(t^2 - t)

    Okay:

    int[e^(t^2 - t)*P)'] = int(e^(t^2 - t)*4t - 2*e^(t^2 - t))

    => e^(t^2 - t)*P = {Since this can't be integrated, I think I've done something wrong }
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  7. #7
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Ideasman View Post
    2t - 1

    Okay, so if P(x) = 2t - 1

    That means M(x) = e^(int(2t - 1)dt) = e^(t^2 - t)

    Multiply M(x) by each term:

    e^(t^2 - t)*P' + e^(t^2 - t)*P = e^(t^2 - t)*4t - 2*e^(t^2 - t)

    Okay:

    int[e^(t^2 - t)*P)'] = int(e^(t^2 - t)*4t - 2*e^(t^2 - t))

    => e^(t^2 - t)*P = {Since this can't be integrated, I think I've done something wrong }
    no, you're correct. new strategy: let's go for separable

    P' + (2t - 1)P = 4t - 2

    \Rightarrow P' = 4t - 2 - (2t - 1)P

    \Rightarrow P' = 2(2t - 1) - (2t - 1)P

    \Rightarrow P' = (2t - 1)(2 - P)

    \Rightarrow \frac {P'}{2 - P} = 2t - 1

    now continue
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  8. #8
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    Quote Originally Posted by Jhevon View Post
    no, you're correct. new strategy: let's go for separable

    P' + (2t - 1)P = 4t - 2

    \Rightarrow P' = 4t - 2 - (2t - 1)P

    \Rightarrow P' = 2(2t - 1) - (2t - 1)P

    \Rightarrow P' = (2t - 1)(2 - P)

    \Rightarrow \frac {P'}{2 - P} = 2t - 1

    now continue
    Ah, tricky algebra manipulation.

    Okay, so from where you left off:

    dP/dt * 1/(2-P) = 2t - 1

    => dP/(2-P) = (2t-1)dt

    Take integral of both sides:

    -ln|P-2| = t^2 - t + C

    Now the second part of the question asked to find the largest interval over which the GENERAL sol'n is defined. This looks defined everywhere, except for when the ln is 0 or negative, but since we have the absolute signs, we don't have to worry about it being neg. Therefore, it's defined (-infinity, 2) or (2, infinity). How am I supposed to know which it's in though.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Ideasman View Post
    Ah, tricky algebra manipulation.

    Okay, so from where you left off:

    dP/dt * 1/(2-P) = 2t - 1

    => dP/(2-P) = (2t-1)dt

    Take integral of both sides:

    -ln|P-2| = t^2 - t + C

    Now the second part of the question asked to find the largest interval over which the GENERAL sol'n is defined. This looks defined everywhere, except for when the ln is 0 or negative, but since we have the absolute signs, we don't have to worry about it being neg. Therefore, it's defined (-infinity, 2) or (2, infinity). How am I supposed to know which it's in though.
    read, again, what i said here regarding this question
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    Quote Originally Posted by Jhevon View Post
    read, again, what i said here regarding this question
    That's the whole point; I don't have a point t_0.
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  11. #11
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Ideasman View Post
    That's the whole point; I don't have a point t_0.
    what is P(x)? and where is it continuous?
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