Simple Differential Equation

  • Thread starter RentonT
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  • #1
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Homework Statement


"Solve this differential equation algebraically, subject to the initial condition that [tex]y=10[/tex] at [tex]t=0[/tex]


Homework Equations


[tex]\frac{dy}{dt} = 2y*\frac{1000-y}{1000}[/tex]


The Attempt at a Solution


I first reduced the right side to [tex]\frac{-y^2}{500} + 2y[/tex]
After that I separated the variables, but I don't know how to integrate that function.
Can anyone point me in the right step?
[tex]\int \frac {dy}{\frac{-y^2}{500}+2y}[/tex]
 
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Answers and Replies

  • #2
Dick
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Have you done any partial fractions recently? You can write the integrand as A/(y)+B/(1000-y). You just need to find A and B.
 
  • #3
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Have you done any partial fractions recently? You can write the integrand as A/(y)+B/(1000-y). You just need to find A and B.
I am in fact working on the section over partial fractions. Thank you very much Dick. I see how partial fractions plays into this. Might I ask how you got [tex]\frac{A}{y} + \frac{B}{1000-y}[/tex]

I'm having trouble seeing the partial fractions. Most of them I have worked with I use the shortcut to Heaviside's Method. Many have been in the form of [tex]\int \frac {Linear}{Quadratic}[/tex]
 
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  • #4
Dick
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I am in fact working on the section over partial fractions. Thank you very much Dick. I see how partial fractions plays into this. Might I ask how you got [tex]\frac{A}{y} + \frac{B}{1000-y}[/tex]

y and (1000-y) were the factors of your original expression before you multiplied it out. You didn't really want to do that.
 
  • #5
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y and (1000-y) were the factors of your original expression before you multiplied it out. You didn't really want to do that.

OK. So what would be on the left-hand side of the equal sign?
[tex]x=\frac{A}{y}+\frac{B}{1000-y}[/tex]

I understand you multiply the left side by one of the denominators on the right side and then plug in the x value that would have zeroed the denominator. I just don't know what goes on the left side. Is it the [tex]\frac{dy}{dt}[/tex]?
 
  • #6
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Nevermind. My poor algebra skills were my downfall this time. I figured out that it's [tex]\int \frac{-500}{y(y-1000)} [/tex]
 

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