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Unformatted text preview: 1.3: Slope Fields and Solution Curves Basic Definitions Consider the differential equation dy dx = f H x , y L . This represents the slope through the point ( x, y ). A direction (or slope) field is generated by drawing a short line segment through each point ( x, y ) having slope given by f H x , y L . Theorem: Existance and Uniqueness of Solutions Let dy dx = f H x , y L , y H a L = b . Suppose f H x , y L and the partial derivatife of f with respect to y , denoted D y f H x , y L = y f H x , y L are both continuous on some rectangle R in the x- y plane containing the point H a , b L . Then, for some open interval I containing a , the initial value problem has a unique solution on I . Example Suppose dy dx =- y . A. Explore the existance and uniqueness theorem. B. Show that y H x L = ce- x is a solution for all x R . Example Suppose dy dx = 2 y , y H L = 0. A. Explore the existance and uniqueness theorem....
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- Fall '11