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Unformatted text preview: Math 334 Lecture #3 2.1: First Order Linear ODEs The first order ODEs dy dt =- 1 5 y + 9 . 8 and ty = 4 t 2- 2 y are examples of the general first order linear ODE dy dt + p ( t ) y = g ( t ) where p ( t ) and g ( t ) are functions continuous on a common open interval I . Method of Solution (Leibniz). The expression on the left-hand side of the differen- tial equation, dy dt + p ( t ) y, is similar to the derivative of the product of y with a differentiable function, say, ( t ): d dt ( t ) y = ( t ) dy dt + ( t ) y. This suggests multiplying the ODE through by ( t ): ( t ) dy dt + ( t ) p ( t ) y = ( t ) g ( t ) . For the left-hand side of this to be [ ( t ) y ] requires that ( t ) satisfies the differential equation ( t ) p ( t ) = ( t ) . [To solve one differential equation we have to solve another differential equation! We will see this idea again and again.] Rewrite the ODE for with all the s on one side: ( t ) ( t ) = p (...
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This note was uploaded on 03/08/2011 for the course MATH 334 taught by Professor Smith during the Spring '11 term at Vanderbilt.
- Spring '11