Is there a unique solution to the given ODE?

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In summary, the given ODE has one solution defined in an open segment containing 0. The "existence and uniqueness" theorem can be used since the function is differentiable in both variables except at (0,0). The initial value of y(0)=5 and y'(0)>0, leading to the conclusion that y'(x)>0 for all x in the open segment. Additionally, the function is always greater than 2, as any max or min must occur at y=2 and the initial value is greater than 2.
  • #1
TheForumLord
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Homework Statement


Given This ODE:

y' = (y-2) (x^2+y)^5
y(0)=5

A. Show that this problem has one solution that is defined in an open segment that contains 0.

B. Let y(x) be a solution for this problem. Prove that y(x)>2 for every x in I and conclude that y'(x)>0 in I.
Hint: You can use the solution of the problem: y'=(y-2)(x^2+y)^5 , y(x0)=2


Help is needed !

TNX!


Homework Equations





The Attempt at a Solution

 
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  • #2


This is of the form y'= f(x,y). Your f, [itex](y-2)(x^2+ y)^5[/itex], is differentiable in both variables in any region that does not include (0,0) so you can use the "existence and uniqueess" theorem.

Further, since y(0)= 5, [itex]y'(0)= (5-2)(0^2+ 5)^5> 0[/itex] and [itex]y'(x)= 0[/itex] only where y= 2 or [itex]y= -x^2[/itex]. The latter is impossible so any max or min must be at y= 2. Since the initial value is 5, the derivative is positive there, and can become negative only at y= 2, the function is always larger than 2.
 
  • #3


Hey there HallsofIvy,
There are some things I didn't understand in your answer:
The initial value is 5 indeed. and from y=5 the function goes up. But how can we know what happened before y=5? Maybe there was a point that was less then y=2? There can be an inflection point in y=2, and then there are values less than y=2...

How can we solve it?


TNX in advance!
 
Last edited:

What is an ODE (ordinary differential equation)?

An ODE is a mathematical equation that relates a function to its derivatives. It involves a single independent variable and one or more dependent variables, and the derivatives describe the rate of change of the dependent variables with respect to the independent variable.

What is a one solution ODE problem?

A one solution ODE problem is an ODE that has a unique solution for a given set of initial conditions. This means that the ODE can be solved to find a specific function that satisfies the equation and the given initial conditions.

How do you solve a one solution ODE problem?

There are various methods for solving one solution ODE problems, such as separation of variables, integrating factors, and substitution. The specific method used depends on the form of the ODE and the initial conditions given.

What are the applications of one solution ODE problems?

One solution ODE problems have a wide range of applications in various fields of science and engineering. They are used to model and analyze physical systems, such as in mechanics, electromagnetics, and fluid dynamics. They are also used in economics, biology, and other areas to study dynamic systems and their behavior over time.

What are some common challenges in solving one solution ODE problems?

Some common challenges in solving one solution ODE problems include identifying the correct method to use, dealing with complex initial conditions, and handling nonlinear or higher-order ODEs. It is also important to check for errors and verify the solution obtained to ensure its accuracy.

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