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**Unformatted text preview: **Math 334 Lecture #36 Â§ 9.2: Autonomous Systems and Stability An autonomous system of two first-order differential equations in x and y is dx/dt = F ( x,y ) , dy/dt = G ( x,y ) where F and G are continuous and have continuous partial derivatives on an open domain D of the xy-plane. The Fundamental and Uniqueness Theorem guarantees that through each initial condi- tion ( x ( t ) ,y ( t )) = ( x ,y ) in D there passes a unique solution x = Ï† ( t ), y = Ïˆ ( t ) of the system defined on an open interval I containing t . Writing ~x = ( x,y ) and f ( ~x ) = ( F ( x,y ) ,G ( x,y )), the above system and initial condition in vector form are ~x = f ( ~x ) , ~x ( t ) = ~x , and its unique solution is ~x ( t ) = ( Ï† ( t ) ,Ïˆ ( t )) . Outcome A: Finding Equilibrium Solutions . The simplest solutions of ~x = f ( ~x ) are the constant vector functions, i.e., equilibrium solutions, if any. Such points ~x are called critical points because they satisfy ~x = f ( ~x ) = 0....

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