Question

Do the solutions to dy/dt = e^y exist for all time? A. Yes, since ey is continuous for all time, the solutions will exist for all time.

Answer 1

Yes, the solutions to dy/dt =
`e^y` exist for all time.

Step-by-step explanation:

The given differential equation is dy/dt =
`e^y`. To determine if the solutions exist for all time, we need to consider the behavior of
`e^y`. Since
`e^y` is continuous for all real numbers, the solutions to the differential equation will also exist for all time. This is because the continuity of
`e^y`ensures that the solutions do not encounter any discontinuities or singularities that would prevent them from existing for all time.

In mathematical terms, the function
`e^y` is defined and continuous for all real numbers y. Therefore, the solutions to dy/dt =
`e^y` are well-defined and continuous for all t, indicating that they exist for all time. This conclusion aligns with the properties of exponential functions and their behavior, ensuring that the solutions to the given differential equation are valid and exist indefinitely.

The continuity of
`e^y` plays a crucial role in establishing the existence of solutions for all time in the context of the given differential equation. This property allows us to confidently assert that the solutions to dy/dt =
`e^y`indeed exist for all time, providing a comprehensive understanding of their behavior and longevity.