Solution of a Differential Equation?

In summary: That makes them "solutions" to the differential equation. Of course, they are not solutions to the "initial value problem" which requires that P(t)= 0 at some particular t- not for all t. The "solutions" are just the horizontal lines P= 0 and P= 30.
  • #1
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Homework Statement


The slope field for the equation [itex]\frac{dP}{dt} = 0.0333P(30−P)[/itex], for [itex]P\geq0[/itex], is shown below.
differential.png

What is the equation of the solution to this differential equation that passes through (0,0)?

The Attempt at a Solution


I'm completely clueless as to how this is to be done. I tried using separable differential equations to solve this, but it didn't give me the right answer as I submitted it, the the website (this question comes from my online homework) said it was wrong. I would like someone to give me a hint as to how to approach this problem, and then I'll proceed with my attempt, thanks.
 
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  • #2
τheory said:

Homework Statement


The slope field for the equation [itex]\frac{dP}{dt} = 0.0333P(30−P)[/itex], for [itex]P\geq0[/itex], is shown below.
View attachment 41087
What is the equation of the solution to this differential equation that passes through (0,0)?

The Attempt at a Solution


I'm completely clueless as to how this is to be done. I tried using separable differential equations to solve this, but it didn't give me the right answer as I submitted it, the the website (this question comes from my online homework) said it was wrong. I would like someone to give me a hint as to how to approach this problem, and then I'll proceed with my attempt, thanks.

Separation of variables should work. It's hard to guess what went wrong unless you show us your work.

[Edit - added] Note the direction field itself gives a large hint.
 
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  • #3
Well I used partial fractions to solve this differential equation and I got this:
[itex]-ln(\frac{|P-30|}{P}) = t + C[/itex]

If I try to solve for C using the initial condition (0,0), then I get that [itex]-ln(\frac{30}{0}) = C[/itex], which is undefined. If C is undefined, then what do you do?
 
  • #4
τheory said:
Well I used partial fractions to solve this differential equation and I got this:
[itex]-ln(\frac{|P-30|}{P}) = t + C[/itex]

If I try to solve for C using the initial condition (0,0), then I get that [itex]-ln(\frac{30}{0}) = C[/itex], which is undefined. If C is undefined, then what do you do?

Yes, you do run into a problem with separation of variables, which is why I added the hint. The problem is that as soon as you divide by P(30-P) you are disallowing the solutions P identically 0 or P identically 30. Yet if you check the original equation you will find that they work. And one of them satisfies your initial conditions. Look at your direction field and see if those don't look like solutions.
 
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  • #5
Wait how am I suppose to generate an equation from looking at the differential field?
 
  • #6
Unfortunately, that slope field is misleading. At (0,0), it appears that the slope is slightly negative but it clearly is not. If P= 0, dP/dt= 0.0333(0)(30- 0)= 0.
Now, what is the simplest function that is 0 at t= 0 and has derivative 0 there? What is its derivative for all t?
 
  • #7
Okay well I tried P = 0 and it worked; I knew this could be an answer but that it couldn't have been this simple. :confused:
 
  • #8
It should be clear that if P(t)= 0 for all t or P(t)= 30 for all t, then the right hand side of
dP/dt= 0.0333P(30- P) is 0. Of course, since those are constant functions so the left side, dP/dt= 0.
 

1. What is a differential equation?

A differential equation is a mathematical equation that describes how a function changes over time or space. It relates the rate of change of a function to the function itself.

2. What is the solution of a differential equation?

The solution of a differential equation is a function that satisfies the equation and its initial conditions. It represents the relationship between the variables in the equation and can be used to predict the behavior of the system described by the equation.

3. How do you solve a differential equation?

There are various methods for solving differential equations, including separation of variables, substitution, and integrating factors. The specific method used depends on the type of differential equation and the initial conditions given.

4. Can all differential equations be solved?

No, not all differential equations can be solved analytically. Some equations may have no closed-form solution, meaning they cannot be expressed in terms of elementary functions. In these cases, numerical methods can be used to approximate the solution.

5. What are the applications of differential equations in science?

Differential equations are used to model and analyze various systems in science, including physical phenomena such as motion, heat transfer, and fluid dynamics. They are also used in fields such as biology, economics, and engineering to describe and predict the behavior of complex systems.

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