Question about multiple functions for a first order ODE

In summary, the conversation discusses finding an implicit solution to a first order ODE by finding a function that satisfies the equation for all values in the domain. It is then questioned whether two different functions that satisfy the ODE can both be correct or if one must be wrong. An example is given where two different functions, H and F, both satisfy the equation but are not equivalent. The conversation also mentions using the uniqueness theorem to determine the correctness of the functions.
  • #1
ArenasField
3
0
The question is as follows:

Suppose you find an implicit solution y(t) to a first order ODE by finding a function H(y, t) such that H(y(t), t) = 0 for all t in the domain. Suppose your friend tries to solve the same ODE and comes up with a different function F(y, t) such that F(y(t), t) = 0 for all t in the domain. Could you both be right, or must one of you be wrong?

I'm pretty lost on this one, honestly. I thought about using the uniqueness theorem (about f(y,t)) being cts and the partial of f(y,t) w.r.t. y), but then I wasn't sure how I could relate this. Can anyone help me out?
 
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  • #2
##y' = 1## is a first order ODE. ## H = y - (t + 4) = 0 ## and ## F = y - (t + 10) = 0 ## both satisfy the ODE, but are different functions. That what you mean ?
 

1. What is a first order ODE?

A first order ODE, or ordinary differential equation, is a type of mathematical equation that involves a function and its derivative. It is typically written in the form dy/dx = f(x,y), where y is the dependent variable, x is the independent variable, and f(x,y) is a function that relates the two variables.

2. What is the difference between a first order ODE and a higher order ODE?

A first order ODE involves only one derivative, while a higher order ODE involves multiple derivatives of the dependent variable. For example, a second order ODE would be written as d^2y/dx^2 = f(x,y).

3. How do you solve a first order ODE?

There are several methods for solving a first order ODE, including separation of variables, integrating factors, and using power series. The specific method used depends on the form of the equation and the initial conditions given.

4. Can a first order ODE have multiple solutions?

Yes, a first order ODE can have multiple solutions. This is because the general solution to a first order ODE includes a constant of integration, which can take on different values for different solutions.

5. What are some real-world applications of first order ODEs?

First order ODEs are used to model a wide range of physical phenomena, including population growth, chemical reactions, and electrical circuits. They are also commonly used in engineering, physics, and economics to understand and predict systems that change over time.

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