For a linear first-order differential equation dy/dt=F(y,t) , we know that the existence and uniqueness theorem tells us that a solution to an initial value problem exists if F is continuous on some interval (because of the intermediate value theorem), and that the solution is unique if the partial derivative with respect to y of F is also continuous on the same interval. However, partial derivatives measure the change of a function in one direction while the other(s) are held constant, which in this case would mean t is held constant while y varies. But if we are talking about an "interval," how can this be? Wouldn't our t interval be restricted? I'm presuming the reasoning behind the uniqueness part has to do with the fact that coplanar curves are said to be parallel and do not intersect, thus meaning only one of a family of coplanar curves can pass through a given point (our initial point). But I'm having trouble understanding how the partial derivative with respect to y relates to that idea.

For a linear first-order ordinary differential equation (ODE) of the form:

$\frac{dy}{dt}=F(y,t)$

where y is a function of t, the existence and uniqueness theorem makes certain guarantees about the solution to this ODE given a particular initial condition. 

Specifically:

• Existence: A solution y(t) to the initial value problem exists if F(y,t) is continuous in both y and t in some neighborhood of the initial condition.
  
  
• Uniqueness: The solution y(t) is unique if, in addition to F being continuous, the partial derivative of F with respect to y, denoted $\frac{\mathrm{\partial }F}{\mathrm{\partial }y}$​, is also continuous in a neighborhood of the initial condition.

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