Question

Question 1 0. NO CALCULATOR IS ALLOWED FOR THIS QUESTION. Submit Show all of your work, even though the question may not explicitly remind you to do so. Clearly label any functions, graphs, tables, or other objects that you use. Justifications require that you give mathematical reasons, and that you verify the needed conditions under which relevant theorems, properties, definitions, or tests are applied. Your work will be scored on the correctness and completeness of your methods as well as your answers. Answers without supporting work will usually not receive credit. Unless otherwise specified, answers (numeric or algebraic) need not be simplified. If your answer is given as a decimal approximation, it should be correct to three places after the decimal point. Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f (x) is a real number. Consider the differential equation = xy?. (a) On the axes provided, sketch a slope field for the given differential equation at the 9 points indicated. OL (b) Find in terms of x and y. Determine the concavity of all solution curves for the given differential equation in Quadrant IV. Give a reason for your answer. (c) Find the particular solution y= f (x) to the given differential equation with initial condition f (2) = -1.

Answer 1

сіх de do sexys dyre, y) = x y} - From Figerre points. At (-0,), 1-1,1)=- AP (0,1 g (0,1)=0 AP (1,1) dy 1,0) = 0 At(0,0) 1 (0,0)=0 Af (1,0) dy (1,0) = 0 At(1,1) dy (-1,-) = 1 Af (0,-) 1 0,-1) = 0 At (1,-1) 18 (1,-1) = -) o means slope is horizontal >o means slope is between o and I To means slope is between II and I CScanned with CamScanner

1) By differentiating a - rys py 9373&y? vy? y = y² + 3 x 245 Тя л очеdstart: oy 40 -) = +3x*y2) y3 <0 solution curves are concane down in I Quadrant c) By variable seperable xy3 z dy=sede By integrating - y = x + c Given f(2)=- =) y = when e=2 =) Jus = C = -5/2. Contribcampartukar solution

simplifying ya = - - - CScanned with CamScanner