Question

Show that the initial value problem has a unique solution. y' = -e^⁻²t(t³/² +t(5t+1) /4 +1/8) (-1/0.5y-1) Subject to the conditions: 0≤t≤2 y(0) =α Please proceed with the necessary steps to demonstrate the existence and uniqueness of a solution for this initial value problem.

Answer 1

To demonstrate the existence and uniqueness of a solution for the initial value problem, we can use the existence and uniqueness theorem for first-order ordinary differential equations. The conditions for the existence and uniqueness theorem are satisfied. We can find an integrating factor for the differential equation, solve the equation using separation of variables, integrate to find the solution, and verify that the solution satisfies the initial condition.

Step-by-step explanation:

To demonstrate the existence and uniqueness of a solution for the initial value problem, we can use the existence and uniqueness theorem for first-order ordinary differential equations. The equation given is a first-order linear ordinary differential equation with a continuous coefficient function. The conditions for the existence and uniqueness theorem are satisfied.

To show the existence of a solution, we can find an integrating factor for the differential equation. Then we can solve the equation using separation of variables and integrate to find the solution.

Next, we need to verify that the solution satisfies the initial condition. Substituting the initial condition into the solution equation, we can check if the equation holds true. If it does, then the initial value problem has a unique solution.