Differential equations usually also contain letters usually x or t that

# Differential equations usually also contain letters

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Differential equations usually also contain letters (usually x or t) that represent numbers, but the solution to a d.e. is a function or a family of functions Exercise 1: In each case verify that the given function is a solution to the given differential equation. a) x 2 dx dy ; 1 x y 2 b) xy’ = 2y ; y = cx 2 (c a constant) c) y’ + 2y = 2x, y = -½e -2x + x - ½ d) y’’ + 9y = 4cos(x); y = ½cos(x) Printed 10/18/2016 8:22 PM -66- cf7d8907366ff4016741d5e1bcff2d7c6af97cbe.doc
Math 124 Calculus 2 3.2 Direction Fields (Section 7.2 in textbook) Recall that the derivative of a function y, which is denoted y’, y’(x) or dx dy , is the slope of the graph of y(x) at any point on the graph. Thus a differential equation, even when it is not solved, can give us information about the slope of the unknown function. A direction field (or slope field) for a differential equation is a graph that has small line segments drawn for various points (x,y) in the plane at regular intervals. Each line segment has a slope that is specified by the differential equation. Exercise 1: a) Complete the table and verify the slope of the corresponding line-segments on the direction field for the differential equation y x dx dy , which is given. x y slope = y x dx dy -1 1 -1 -1 -1 -2 0 0 0 1 0 -1 1 2 1 1 1 0 2 -2 b) Show that y = x – 1 + ce -x is a solution to the differential equation for any constant c. c) Sketch the graphs of the solutions that satisfy the given initial conditions and find the constant c associated with each initial condition: i) y(0) = 1 ii) y(0) = 0 iii) y(0) = -1 Exercise 2: Match each direction field with the corresponding differential equation. i) y 2 x ' y ii) ) y x cos( ' y iii) 2 y 1 ' y iv) xy ' y Printed 10/18/2016 8:22 PM -67- cf7d8907366ff4016741d5e1bcff2d7c6af97cbe.doc
Math 124 Calculus 2 For each differential equation, sketch the solutions that pass through each of (0,0), (0,-1) and (0,1). 3.3 Euler’s Method Exercise 1: Consider the differential equation y’ = y 2 x 2 . It is very difficult to solve this d.e. algebraically, but we can come up with approximate solutions. a) On the direction field shown, sketch the solution that goes through the point (1,1). Printed 10/18/2016 8:22 PM -68- cf7d8907366ff4016741d5e1bcff2d7c6af97cbe.doc
Math 124 Calculus 2 b) Using Euler’s method , construct a table of numerical estimates to the solution of y’ = y 2 – x 2 that goes through the point (1,1). Let h = 0.25 and go from x = 1 to x = 2. Graph the estimates. x y slope = y’ = y 2 – x 2 y = (slope) x 1 1 Exercise 2: (similar to #7 pg. 520 textbook) The slope field for the differential equation 2 xy 2 ' y is given. a) On the slope field sketch the solution having initial condition y(0) = 1. b) Use Euler’s method with step size 0.1 to estimate y(0.6) from the initial value y(0) = 1. x y slope= y’ y = (slope) x Printed 10/18/2016 8:22 PM -69- cf7d8907366ff4016741d5e1bcff2d7c6af97cbe.doc
Math 124 Calculus 2 3.4 Separable Equations A separable differential equation is a first-order differential equation in which y’ can be factored into a function of x multiplied by a function of y.
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