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# L22 - Fall 2003 Math 308/501502 2 First-Order Differential...

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Fall 2003 Math 308/501–502 2 First-Order Differential Equations 2.2 Separable Equations Mon, 08/Sep c 2003, Art Belmonte Summary Separable first-order equations have the form dy dt = g ( t ) h ( y ) or dy dt = g ( t ) f ( y ) . In either case, separate the expressions and differentials to obtain an equation of differential forms; e.g., h ( y ) dy = g ( t ) dt . Then integrate (antidifferentiate) each side, h ( y ) dy = g ( t ) dt , to obtain H ( y ) = G ( t ) + C . This result implicitly defines y in terms of t ; i.e., we have an implicit solution . If we can subsequently solve for y (i.e., if H has an inverse function H - 1 ), then we can write the dependent variable y explicitly as a function of the independent variable t ; i.e., we have an explicit solution . If we have an explicit solution, it may be graphed in MATLAB using the same techniques described in Chapter 3 of your lab manual (or via dfield7 ). If we have an implicit solution, these techniques do not work. MATLAB’s contour command, however, serves admirably! Hand Examples Example A Find the general solution of y 0 = ( 1 + y 2 ) e x . If possible, find an explicit solution.

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L22 - Fall 2003 Math 308/501502 2 First-Order Differential...

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