Lesson 9a - Integrating Factors

Ifthisfailsthenwedonotexpectyoutogoanyfurther

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Unformatted text preview: , y ) Py ( x , y )) P ( x, y ) dy I ( y) Whichever one works out to being a function of only one variable will be the one you try. This will make the equation exact and you can solve it using the exact method. If this fails, then we do not expect you to go any further. Example 1 Show that the following ODE is not exact. Find an integrating factor and show that it is exact when you multiply by the integrating factor. x 2 y 2 xyy' 0 First we show that it is not exact Py 2 y Qx y They are not equal, so this is not an exact ODE. We first try: I ( x) e I ( x) e ( Py ( x , y ) Qx ( x , y )) Q( x, y ) (2 y y) dx xy I ( x) e I ( x) e y dx xy 1 dx x dx I ( x) e ln| x| I ( x) x This makes our new ODE, and then we show it is exact: x 3 xy 2 x 2 yy ' 0 Py 2 xy Qx 2 xy Example 2 Show that the following ODE is not exact. Find an integrating factor and show that it is exact when you multiply by the integrating factor. Sin( x)Tan( y ) (2 Sin( y ) Cos ( x)) y ' 0 First we show that it is not exact Py Sin( x) Sec 2 ( y ) Qx Sin( x) They are not equal, so this is not an exact ODE. We first try: I ( x) e I ( x) e ( Py ( x , y ) Qx ( x , y )) Q( x, y ) dx ( Sin ( x ) Sec 2 ( y ) Sin ( x )) dx 2 Sin ( y ) Cos ( x ) This will not simplify to something that works out in terms of x alone… so let us try the other I(y): I ( y) e ( Sin ( x ) Sin ( x ) Sec 2 ( y )) dy Sin ( x )Tan ( y ) I ( y) e Sin ( x )(1 Sec 2 ( y )) dy Sin ( x )Tan ( y ) Example 2 Continued I ( y) e I ( y) e (1 Sec 2 ( y )) dy Tan ( y ) ( Tan 2 ( y )) dy Tan ( y ) I ( y ) e Tan ( y )dy I ( y ) e ln|cos( y )| I ( y ) cos( y ) This makes our new ODE, and then we show it is exact: Sin( x)Tan( y )Cos ( y ) (2Sin( y )Cos ( y ) Cos ( x)Cos ( y )) y ' 0 Sin( y ) Sin( x) Cos ( y ) (2Sin( y )Cos ( y ) Cos ( x)Cos ( y )) y ' 0 Cos ( y ) Example 2 Continued Sin( x) Sin( y ) (2 Sin( y )Cos ( y ) Cos ( x)Cos ( y )) y ' 0 Py Sin( x)Cos ( y ) Qx Sin( x)Cos ( y )...
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This note was uploaded on 02/07/2014 for the course MATH 1004 taught by Professor Mark during the Summer '00 term at Carleton CA.

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