3. (10 marks). Solutions to differential equations can be approximated numerically. One of the simplest methods for generating numerical approximations is called Euler's Method (see Section 19.3 for more details and worked examples). The solution to the first-order differential equation y' = f(z, y), y(To) = yo at the point r = To is approximated using a difference quotient as follows: y(To + h) - y(TO) = f(ro, y(zo)) h We can rearrange this equation to obtain y(roth) = y(ro) + hf(ro, y(ro)) For reasonable choices of h, this allows us to find an approximation to the solution at 1 = To + h if we know the solution at To. y(z) = y(roth) = y(zo) + hf(zo, y(zo)) This procedure is iterative, so once we know an approximation of the solution at T1 = To + h, we can find an approximation of the solution at z2 = rith = to + 2h. y(21 ) = y(zo + h) = y(ro) + hf(ro, y(10)) y(x2) = y(r th) = y(m) + hf(my(z)) y(13) = y(z, th) = y(x2) + hf(x2, y(32)) Consider the following initial value problem. y' = -Ary, y(0) = 1 (a) Solve the initial value problem explicitly. (b) Use Euler's Method to approximate the solution y(1) using h = 0.2. (c) Use Desmos to plot your answer to part (a) along with each step of your approximation to part (b). (d) Compare your approximation of y(1) from part (b) to the exact value of y(1) from part (a) by calculating percent error: (true value - approximation)/ (true value)