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Unformatted text preview: MA 36600 LECTURE NOTES: WEDNESDAY, JANUARY 21 First Order Linear Equations Definitions. So far we have considered initial value problems in the form dy dt =- b + ay, y (0) = y where a and b are constants. We showed that the solution is y ( t ) = b a + C e at where C = y- b a . In general, an ordinary differential equation is said to be a first order equation if it is a relation in the form F ( t, y, y ) = 0 . Upon expressing the derivative y in terms of the other variables, we find an expression in the form dy dt = G ( t,y ) for some function G ( t,y ). We say that a function y = y ( t ) is a solution if it satisfies this equation. If y ( t ) = y is an initial value, we think of the graph of y = y ( t ) as passing through the point ( t ,y ). We say that this first order equation is linear if G ( t,y ) = g ( t )- p ( t ) y for some functions g ( t ) and p ( t ). That is, if the differential equation is in the form dy dt + p ( t ) y = g ( t ) . Integrating Factors. Consider the first order linear differential equation dy dt + p ( t ) y = g ( t ) . Choose a function = ( t ). Multiply the first order differential equation above by this function: ( t ) dy dt + ( t ) p ( t ) y = ( t ) g ( t ) . Say that we can choose a function = ( t ) such that i. ( t ) is not identically zero, and ii. ( t ) = p ( t ) ( t )....
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