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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 functionsExercise 1:In each case verify that the given function is a solution to the given differential equation.a) x2dxdy; 1xy2b) xy’ = 2y ; y = cx2(c a constant)c)y’ + 2y = 2x, y = -½e-2x + x - ½d) y’’ + 9y = 4cos(x); y = ½cos(x) Printed 10/18/2016 8:22 PM -66-cf7d8907366ff4016741d5e1bcff2d7c6af97cbe.doc
Math 124Calculus 23.2Direction Fields (Section 7.2 in textbook)Recall that the derivative of a function y, which is denoted y’, y’(x) ordxdy, is the slopeof 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 correspondingline-segments on the direction field for the differential equation yxdxdy, which is given.xyslope = yxdxdy-11-1-1-1-200010-11211102-2b)Show that y = x – 1 + ce-xis 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) = 1ii) y(0) = 0iii) y(0) = -1Exercise 2:Match each direction field with the corresponding differential equation.i) y2x'yii) )yxcos('yiii) 2y1'yiv) xy'yPrinted 10/18/2016 8:22 PM -67-cf7d8907366ff4016741d5e1bcff2d7c6af97cbe.doc
Math 124Calculus 2For each differential equation, sketch thesolutions that pass through each of (0,0),(0,-1) and (0,1).3.3Euler’s MethodExercise 1:Consider the differential equation y’ = y2–x2. It is very difficult to solve this d.e. algebraically, butwe can come up with approximate solutions. a)On the direction field shown, sketch the solutionthat goes through the point (1,1).Printed 10/18/2016 8:22 PM -68-cf7d8907366ff4016741d5e1bcff2d7c6af97cbe.doc
Math 124Calculus 2b) Using Euler’s method, construct a table of numerical estimates to the solution of y’ = y2– x2that goes through the point (1,1). Let h = 0.25 and go from x = 1 to x = 2. Graph the estimates.xyslope = y’ = y2– x2y = (slope) x11Exercise 2: (similar to #7pg. 520 textbook) Theslope field for thedifferential equation2xy2'yis given.a)On the slope field sketch the solution havinginitial condition y(0) = 1.b)Use Euler’s method with step size 0.1 toestimate y(0.6) from the initial value y(0) =1.xyslope= y’y = (slope) xPrinted 10/18/2016 8:22 PM -69-cf7d8907366ff4016741d5e1bcff2d7c6af97cbe.doc
Math 124Calculus 23.4SeparableEquationsA separabledifferential equation is a first-order differential equation in which y’ can be factored into a function of x multiplied by a function of y.