7-1 Differential equations & Slope fields

Def: Differential Equation: an equation that involves a derivative The order of a differential equation is order of highest deriv. involved Ex 1) Find all functions y that satisfy Solution is antiderivative  any form of: general solution *What if we are looking for a particular solution? Need an initial condition to solve for C

Ex 2) Find the particular solution to the equation whose graph passes through the point (1, 0). General solution: Ex 3) (Handling Discontinuity in an Initial Value Problem) Find the particular solution to the equation whose graph passes through (0, 3).

Ex 4) (Using Fund Thm to Solve an Initial Value Prob) Find the solution to the differential equation for which f (7) = 3. Check: Works!  Ex 5) Graph the family of functions that solve the differential equation y = sin x + C Graph these  {–3, –2, –1, 0, 1, 2, 3}  L1 Y1 = sin x + L1  Zoom TRIG

Ex 6) Construct a slope field for the differential equation Slope Fields What if we want to reproduce a family of graphs without solving a differential equation? (Which is sometimes really hard to do.) We can, by examining really small pieces of the “slopes” of the differential curve and repeating this process many times over. The result is called a slope field. Ex 6) Construct a slope field for the differential equation Think about y values at certain x value spots… When x = 0, cos x = 1 When x =  or x = –, cos x = –1 When x = (odd mult) , cos x = 0 When x = 2 or x = –2, cos x = 1 2 –2 4 –4

*Note: You can graph the general solution with a slope field even if you cannot find it analytically

Ex 7) Use a calculator to construct a slope field for the differential equation and sketch a graph of the particular solution that passes through the point (2, 0). Slopes = 0 along x + y = 0 Slopes = –1 along x + y = –1 Slopes get steeper as x inc Slopes get steeper as y inc Draw a smooth curve thru point (2, 0) that follows slopes in slope field

Horizontal when x = 0 & Vertical when y = 0 Ex 8) Use slope analysis to match each of the following differential equations with one of the slope fields pictured. Zero slope along x – y = 0 B Zero slope along both axes D Horizontal when x = 0 & Vertical when y = 0 A Vertical when x = 0 & Horizontal when y = 0 C

homework Pg. 331 #1 – 46 (skip mult of 3), 50, 59