Difficult-to-solve differential equations can always be approximated by numerical methods. We look at one numerical method called Euler's Method. Euler's method uses the readily available slope information to start from the point (xo, yo) then move from one point to the next along the polygon approximation of the graph of the particular differential equation to ultimately reach the terminal point, (x,y). Al- though interested in determining all of the points along the differential equation, it is often the case that the value of y, at the terminal point is of most interest. More specifically, let y = f(x) be the solution to the differential equation = g(x,y), with y(x) = yo For Xo ≤x≤x, and let X1+1=x+h, where h = -x and 12 Yi+1=Y₁+ g(x,y)h For 0≤i≤n-1, then f(x1+1) ≈Yi+1 g(x,y)=... 1 1. Approximate: x = y - 2x start at (x, y) = (0,1), 0 ≤ x ≤2, when h = 0.4 Since dy = y - 2x then dx (i) y - 2x (iii) 3e¹/2 (ii) 2x+2-ex And since x = 0, yo = 1 then g(x,yo) yo - 2x₁ = 1-2(0)= (i) 0 (ii) 1 (iii) 2, And since h = 0.4, (i) 1 Y₁ yo+g(xo, yo)h = 1 + 1(0.4)= But now, since x₁ = x0 + h = 0 + 0.4 0.4 and g(x₁, y₁) = y₁- 2x₁ = 1.4 - 2(0.4)= (ii) 1.4 (iii) 1.8, (i) 0.6 And since h = 0.4, (ii) 0.8 (iii) 1.0, (i) 1.64 y2y+g(x1,y₁)h = 1.4 + 0.6(0.4) = Complete the following table: i Euler's Approximation (ii) 1.84 (iii) 2.04 Actual Solution Difference y-f(x) 0 1 2 3 4 5 a) Repeat question 1 with h = 0.1 b) Euler's method improves or worsens for smaller h subintervals. 3 Approximate bear population assuming the bear population grows according to following differential equation: dy at=y(y+1)(y+3) Assume initial population at time t = 0 is y = 5, use Euler's method, where h = 1 year, to approximate bear population at time t = 3 years. NB: Follow the same steps as question one