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Unformatted text preview: Math 220  Review sheet for Exam 1 First order equations Separable equations can be written in the form f ( y ) dy/dt = g ( t ), with solution Z f ( y ) dy = Z g ( t ) dt + C. Linear equations can be written in the form y + p ( t ) y = g ( t ). To solve, put ( t ) = e R p ( t ) dt . Multiply through by to get d dt ( y ) = g, Integrate and divide by to get y ( t ) =  1 ( t ) Z t ( s ) g ( s ) ds + C 1 ( t ) . Modeling problems examined include exponential growth and decay, logistic growth, New tons law of cooling, simple problems with Newtons 2nd law, including computing the escape velocity. Autonomous ODEs have the form dy/dt = f ( y ); the right hand side is independent of t . Complete qualitative information about the solutions can be obtained by finding the equilibrium points (those values of y at which f ( y ) = 0) and determining their stability (use the sign of f to determine if nearby points are attracted or repelled Direction fields: For the ODE y = f ( t,y ), draw a small line segment at the point ( t,y ) with slope f ( t,y ). The field of line segments is called the direction field for the ODE....
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 Spring '08
 LERNER
 Differential Equations, Linear Equations, Equations, Derivative, Ode, Partial differential equation, homogeneous equation, fundamental set

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