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Math Help - Separable differential equations problems

  1. #1
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    Separable differential equations problems

    Two problems I'm stuck on.( sorry these aren't separable, they are first order linear, i wrote the wrong thing)

    1: t*(dy/dt) + 2y = t^2 - t + 1

    2: t*(dy/dt) + (t+1)*y = t


    Any help finding the general solutions would be much appreciated.
    I already got the integrating factors as 1: exp(2lnt) and 2: exp(t+lnt)
    Last edited by Obstacle1; August 13th 2007 at 04:35 AM.
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  2. #2
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    For #1:

    ty'+2y=t^{2}-t+1

    Divide by t: y'+\frac{2}{t}y=t-1+\frac{1}{t}

    The integrating factor is 2/t.

    e^{\int{\frac{2}{t}}dt}=t^{2}

    \frac{d}{dt}[t^{2}y]=t^{2}(t-1+\frac{1}{t})=t^{3}-t^{2}+t

    Integrate:

    t^{2}y=\frac{t^{4}}{4}-\frac{t^{3}}{3}+\frac{t^{2}}{2}+C

    Solve for y:

    y=\frac{t^{2}}{4}-\frac{t}{3}+\frac{1}{2}+Ct^{-2}
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  3. #3
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    Quote Originally Posted by Obstacle1 View Post
    Two problems I'm stuck on.( sorry these aren't separable, they are first order linear, i wrote the wrong thing)

    1: t*(dy/dt) + 2y = t^2 - t + 1

    2: t*(dy/dt) + (t+1)*y = t


    Any help finding the general solutions would be much appreciated.
    I already got the integrating factors as 1: exp(2lnt) and 2: exp(t+lnt)
    When you have a linear ODE like y'+P(x)y=Q(x) the integrating factor is given by \color{blue}\mu(x)=\exp\int P(x)\,dx

    Then you amplify the equation by \mu to set the left side into a product's function derivative, finally all you have to do is integrate.
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  4. #4
    Math Engineering Student
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    Quote Originally Posted by Obstacle1 View Post
    I already got the integrating factors as 2: exp(t+lnt)
    \exp(t+\ln t)=\exp(\ln e^t+\ln t)=t\cdot e^t

    That's all.
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