Given the initial value problem y' : 10 e y, y(0) : 1, use Euler's method with h :% to estimate the value of y(1). (Give your answer correct to at least three decimal places.) Using separation of variables, solve the initial value problem y' : 10 e y, y(0) : 1. Use this solution to find the exact value of y(1). (Give your answer correct to at least three decimal places.) Aside: Compare the two answers. Submll Answer ' 2. [l9 Points] DETAILS MY NOTES This question concerns the differential equation $1? : r e y. Here is the slope field for this differential equation. You will need to print out a copy of this slope field. (Note that the graph uses x and y instead of t and y.) ////.r//// u \ \ \\\\'\\\\\\\\\\\\\\\\\ '\\\\\\\\\\\\\\\\\\\\\ \\\\\\\\\\\\\\\\ \\\\\\\\\\\\\\\\\ \\\\~-\\\\\\\\\\\\\\\\ (a) The slope field suggests that there is a straight line solution of this differential equation. Find the equation of this straight line (your answer should be given in terms of t). y: (b) In this part of the question, you will show that the function y : rm : 29" + i: e 1 is a solution of the initial value problem given by the above differential equation with the initial condition y(0) : 1. Calculate % by differentiating the formula for f(t). d dt Calculate t a y by substituting for y. tey: Simplify your answers. This shows that in : t e y. Finally, show that the fundion satises the initial condition by calculating f(0). K0) : (c) On your copy of the slope field, sketch the solution curve of the differential equation that passes through the point (0, 2). Based on the slope eld, can this solution pass through the point (0.5, 2.2)? 0 Yes O No SubmllAnswer ' 3. [/7 Points] DETAILS MY NOTES This question concerns the differential equation 41? : 2y(1 r y). Here is the slope field for this differential equation. You will need to print out a copy of this slope field. (Note that the graph uses x and y instead of t and y.) \ \ \ \ \ \ \ \ \ / / / / / / / K \ \ \ \ \ \ \ \ "WM/H \\\\\\\\\ WWI/f" (a) with the help of the figure or by considering the differential equation, how many constant solutions does this DE have? S constant solution(s) (b) Give the constant solution(s). If there is more than one enter your answers as a commarseparated list. y: (c) If y0 > 1, is the solution of the initial value problem given by the differential equation above together with the initial condition y(o) : yo decreasing for all r > 0? (Notice that you can answer this question either by examining the slope field or by considering the sign of "1 when y is larger than 1.) dt 0 Yes n Mn