Maximal solution to ODE (HEEELP)

[Susanne217]

Homework Statement

Given the ODE

$$\frac{dy}{dt} = y \cdot t^{-1} -2y^2$$

Find all the maximal solutions defined on the interval $$(0,\infty) \times \mathbb{R}.$$

Homework Equations

The Attempt at a Solution

This looks like a Bernoulii equation I find the general solution to be be

$$y(t) = \frac{t}{t^2+k}$$ but as I see its not possible to construct any other solution for this equation than the one above. So what I don't get is how its possible to say that there is more than one maximal solution of the equation above?

Sincerely
Susanne.

 

[mighty2000]

Hello Susanne,

Thank you for your post. It seems that you have correctly found the general solution to the given ODE. However, there can be more than one maximal solution to an ODE, depending on the initial conditions. A maximal solution is defined as the largest possible solution that satisfies the given initial conditions. So, for different initial conditions, there can be different maximal solutions.

In this case, the maximal solutions on the interval (0,\infty) \times \mathbb{R} are determined by the initial condition y(0) = y_0, where y_0 is any real number. So, for each different value of y_0, we can have a different maximal solution. For example, if y_0 = 1, then the maximal solution is y(t) = \frac{t}{t^2+1}. But if y_0 = 2, then the maximal solution becomes y(t) = \frac{t}{t^2+2}. Both of these solutions are valid and satisfy the given ODE, but they are different because of the different initial conditions.

I hope this helps to clarify why there can be more than one maximal solution to an ODE. Keep up the good work with your studies!



Scientist