Question

dy dt = (1 t)|y| 1 2 y(1) = 1 for each of the following initial value problems state whether or not the existence and uniqueness theoremguarantees that a solution exists.

Answer 1

The existence and uniqueness theorem guarantees that a solution exists for this initial value problem.

Step-by-step explanation:

The existence and uniqueness theorem states that for a first-order ordinary differential equation (ODE) y' = f(t,y) with initial condition y(1) = 1, a unique solution exists in a neighborhood of the initial condition if f(t,y) is continuous and satisfies the Lipschitz condition with respect to y.

In this case, we have the ODE dy/dt = (1 - t)|y|^(1/2) with the initial condition y(1) = 1. We can see that the function f(t,y) = (1 - t)|y|^(1/2) is continuous and satisfies the Lipschitz condition with respect to y in a neighborhood of the initial condition. Therefore, the existence and uniqueness theorem guarantees that a solution exists for this initial value problem.