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Unformatted text preview: 8 Slope Fields: Graphing Solutions Without the Solutions Up to now, our efforts have been directed mainly towards finding formulas or equations describing solutions to given differential equations. Then, sometimes, we sketched the graphs of these solutions using those formulas or equations. In this chapter, we will do something quite different. Instead of solving the differential equations, we will use the differential equations, directly, to sketch the graphs of their solutions. No other formulas or equations describing the solutions will be needed. The graphic techniques and underlying ideas that will be developed here are, naturally, especially useful when dealing with differential equations that can not be readily solved using the methods already discussed. But these methods can be valuable even when we can solve a given differential equation since they yield pictures describing the general behavior of the possible solutions. Sometimes, these pictures can be even more enlightening than formulas for the solutions. 8.1 Motivation and Basic Concepts Suppose we have a first-order differential equation that, for motivational purposes, just cannot be solved using the methods already discussed. For illustrative purposes, lets pretend 16 dy dx + xy 2 = 9 x is that differential equation. (True, this is really a simple separable differential equation. But it is also a good differential equation for illustrating the ideas being developed.) For our purposes, we need to algebraically solve the differential equation to get it into the derivative formula form, y = F ( x , y ) . Doing so with the above differential equation, we get dy dx = x 16 ( 9 y 2 ) . (8.1) Remember, there are infinitely many particular solutions (with different particular solutions typically corresponding to different values for the general solutions arbitrary constant). Lets now pick some point in the plane, say, ( x , y ) = ( 1 , 2 ) , let y = y ( x ) be the particular solution 157 158 Slope Fields to the differential equation whose graph passes through that point, and consider sketching a short line tangent to this graph at this point. From elementary calculus, we know the slope of this tangent line is given by the derivative of y = y ( x ) at that point. And fortunately, equation (8.1) gives us a formula for computing this very derivative without the bother of actually solving for y ( x ) ! So, for the graph of this particular y ( x ) , Slope of the tangent line at ( 1 , 2 ) = dy dx at ( x , y ) = ( 1 , 2 ) = x 16 ( 9 y 2 ) at ( x , y ) = ( 1 , 2 ) = 1 16 ( 9 2 2 ) = 5 16 . Thus, if we draw a short line with slope 5 / 16 through the point ( 1 , 2 ) , that line will be tangent at that point to the graph of a solution to our differential equation....
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