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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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