If the function f(x,y) is continuous near the point (a,b), then at least one solution of the differential equation y' = f(x,y) exists on some open interval I containing the point x = a and,
moreover, that if in addition the partial derivative
$\frac{\partial f}{\partial y}$
is continuous near (a,b) then this solution is unique on some (perhaps smaller) interval J. Determine whether existence of at least
one solution of the given initial value problem is thereby guaranteed and, if so, whether uniqueness of that solution is guaranteed.
$\frac{dy}{dx}$ = 9x³y³; y(8) = -3
The theorem implies the existence of at least one solution because f(x,y) is continuous near the point. This solution is unique because
$\frac{\partial f}{\partial y}$
is also continuous near
that same point.
The theorem implies the existence of at least one solution because f(x,y) is continuous near the point
However, this solution is not necessarily unique because
$\frac{\partial f}{\partial y}$
is not continuous near that same point.
