Finding the limit and a differential equation

In summary, the conversation is about finding the limit of a solution curve for the differential equation dy/dx= y-2y^2 containing the point (0, 0.25). The options for the limit are no limit, 0, 0.25, 0.5, and 2. The person is having trouble finding the solution curve, as it requires the integral of 1/(y-2y^2). They are advised to use partial fractions. The solution curve is found to be y= 1/(e^-x + 2) +C but with a different value for C. The question then becomes whether the limit depends on the value of C or not.
  • #1
cokezero
11
0
i can't seem to figure this out...

if the differential equation dy/dx= y-2y^2 has a solution curve y=f(x) contianing point (0, 0.25) , then the limit as x approaches infinity of f(x) is



a)no limit

b. 0

c. 0.25

d. 0.5

e. 2


i usually just separate the variables and find f(x) then take the limit, but i can't seem to find f(x) b/c it would require the integral of 1/(y-2y^2)
 
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  • #2
cokezero said:
i usually just separate the variables and find f(x) then take the limit, but i can't seem to find f(x) b/c it would require the integral of 1/(y-2y^2)

so, integrate [itex]\frac{1}{y-2y^2}[/itex]! Partial fractions will do it.
 
  • #3
yeah i know...
i get the equation

y= 1/(e^-x + 2) +C
without the c value it is 1/2 for the limit
but with the c value which is -1/12 i get a limit of 5/12 which is not an answer choice...

so the question becomes, does the limit depend on the c value or not?
 
  • #4
That's not the answer I get for y(x). Try checking your work again. If you still can't figure it out, post what you've done and I'll try to tell you what's wrong!
 

1. What is the limit of a function?

The limit of a function is the value that the function approaches as the input approaches a certain value. It is often denoted by the notation "lim x→a f(x)", which means the limit of f(x) as x approaches a.

2. How do you find the limit of a function?

To find the limit of a function, you can use the algebraic limit laws or graphing techniques. For algebraic limit laws, you can factor, simplify, or use L'Hôpital's rule. For graphing techniques, you can use a graphing calculator or plot the function to visually determine the limit.

3. What is a differential equation?

A differential equation is an equation that relates a function with its derivatives. It describes how a function changes over time and is often used to model real-world phenomena in fields such as physics, engineering, and economics.

4. How do you solve a differential equation?

The method for solving a differential equation depends on the type of equation. Some common techniques include separation of variables, integrating factors, and using substitution or power series methods. The solution may also involve finding a particular solution for a given initial condition.

5. Why is finding the limit and solving a differential equation important?

Finding the limit of a function is important in calculus because it helps determine the behavior of a function at a specific point and is used in various calculus concepts such as continuity, derivatives, and integrals. Solving differential equations is crucial in many fields of science and engineering as it allows us to make predictions and analyze the behavior of complex systems.

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