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2.1 - Math 334 Lecture#3 2.1 First Order Linear ODEs The...

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Math 334 Lecture #3 § 2.1: First Order Linear ODE’s The first order ODE’s 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, μ ( t ): d dt μ ( t ) y = μ ( t ) dy dt + μ ( t ) y. This suggests multiplying the ODE through by μ ( t ): μ ( t ) dy dt + μ ( t ) p ( t ) y = μ ( t ) g ( t ) . For the left-hand side of this to be [ μ ( t ) y ] requires that μ ( t ) satisfies the differential equation μ ( t ) p ( t ) = μ ( 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 μ with all the μ ’s on one side: μ ( t ) μ ( t ) = p ( t ) . The left-hand side is the derivative of a natural log: d dt ln | μ ( t ) | = p ( t ).

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2.1 - Math 334 Lecture#3 2.1 First Order Linear ODEs The...

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