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Chapter1Section3 - 1.3 Slope Fields and Solution Curves...

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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 0 L = 0. A. Explore the existance and uniqueness theorem. B. Show that y 1 H x L = 0 and y 2 H x L = x 2 are solutions.
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