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Thread: ODE Autonomous Help

  1. #1
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    ODE Autonomous Help

    So I have a problem that, with help, I figured out the solution to part A&B. I overlooked a part C.

    So the differential equation is (dy/dx) = (y+4)(y-5)

    And the question is:

    Solve the differential equation in problem 1 above for y(x) with y(0) = 1, and find lim y(x) as x approaches infinity. Would you just distribute (y+4)(y-5) and integrate, then plug in the initial condition? I drew a slope field with solutions curves for general solutions in Part B, and they also ask does this limit agree with the pictures in question 1? It is a 1 which I found to be approaching a horizontal asymptote. I'm guessing thats what they are asking to confirm?
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  2. #2
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    Re: ODE Autonomous Help

    No, the equation is separable.

    $\displaystyle \begin{align*} \frac{\mathrm{d}y}{\mathrm{d}x} &= \left( y + 4 \right) \left( y - 5 \right) \\ \frac{1}{\left( y + 4 \right) \left( y - 5 \right) }\,\frac{\mathrm{d}y}{\mathrm{d}x} &= 1 \\ \int{ \frac{1}{\left( y + 4 \right) \left( y - 5 \right) } \,\frac{\mathrm{d}y}{\mathrm{d}x}\,\mathrm{d}x} &= \int{ 1\,\mathrm{d}x} \\ \int{ \frac{1}{\left( y + 4 \right) \left( y - 5 \right) } \,\mathrm{d}y} &= \int{ 1\,\mathrm{d}x} \end{align*}$

    Go from here. You can evaluate the left hand integral using Partial Fractions.
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  3. #3
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    Re: ODE Autonomous Help

    Actually, you can answer this question without explicitly solving the differential equation. First, since dy/dx= (y+5)(y-4)= 0 for y= -5 and y= 4, y(x)= -5 for all x, and y(x)= 4x, for all x are "constant" solutions. Further, y= 1 lies between -5 and 4, dy/dx= (1+ 5)(1- 4)= 6(-3)= -18 so the y is decreasing at that point, and only can change sign at places where dy/dx= 0, that is, when y= -5 and 4. Finally, two solution curves cannot cross so the function, y(x), must decrease down to -5 as x goes to infinity.
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    Re: ODE Autonomous Help

    Quote Originally Posted by HallsofIvy View Post
    Actually, you can answer this question without explicitly solving the differential equation.
    Except the question says to solve the DE...
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  5. #5
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    Re: ODE Autonomous Help

    You are right. What I said was probably relevant to parts A and B! Also, ProveIt, the equations in your 'signature' don't show up properly. You might want to change the [tex] in it.
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