Question

We know that y(x)=0 is a solution of the following ODE dx/dy =3xy 3/2 satisfying the initial condition y(244)=0. Are there any other solutions with y(244)=0 ? Explain why or why not. Can there be solutions with y(244)=−3 ?

Answer 1

1. No, there are no other solutions with
`\(y(244) = 0\)` for the given ODE; uniqueness theorem implies uniqueness for a given initial condition.

2. Yes, there can be other solutions with
`\(y(244) = -3\)` for the ODE; uniqueness theorem doesn't guarantee uniqueness for different initial conditions.

Step-by-step explanation:

1. **Main Answer for Part (a):** There are no other solutions with
`\(y(244) = 0\)` for the given ordinary differential equation (ODE)
`\( (dx)/(dy) = 3xy^(3/2) \)` that satisfy the initial condition
`\(y(244) = 0)`.

2. **Explanation for Part (a):** The uniqueness theorem for first-order ODEs states that if a solution to an initial value problem exists, it is unique in a certain interval around the initial point. In this case, since
`\(y(x) = 0\)` is a solution satisfying
`\(y(244) = 0\)`, and assuming the conditions of the uniqueness theorem are met, there are no other solutions with the same initial condition.

3. **Main Answer for Part (b):** There can be other solutions with
`\(y(244) = -3\)` for the given ODE
`\( (dx)/(dy) = 3xy^(3/2) \)`.

4. **Explanation for Part (b):** The uniqueness theorem doesn't guarantee uniqueness for different initial conditions. Therefore, there can be multiple solutions with different initial conditions. To find solutions with
`\(y(244) = -3\)`, one would need to solve the ODE with this new initial condition, and it's possible that such solutions exist.