Consider the initial value problems IVPs given below where t

Consider the initial value problems (IVPs) given below where the differential equation is written in the normal form y\' = f(x,y). For each IVP, respond to the following: Is the IVP guaranteed to have a solution? If so, is it guaranteed to be unique?

Solution

An IVP can have no solution , unique solution or infinitely many solutions.

1. IVP y\'² +y² +1=0, y(0)=1 has no solution.

2. IVP y\'=2x, y(0)=1 has unique solution y=x²+1.

3. IVP xy\'=y-1 , y(0)=1 has infinitely many solutions. General solution is y=cx+1. using initial condition y(0)=1, we get c to be arbitrary. Thus this IVP has infinitely many solution.

However depending upon certain conditions on IVP, we have existence and uniqueness of the solutions.