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bao gồm các phương pháp để tìm diện tích xấp xỉ theo đồ thị của functionflx) giữa hai giá trị x. Đơn giản nhất trong số này sử dụng quy tắc hình thang. Nếu chúng tôi chia diện tích dưới đường cong thành một số lượng đủ lớn các bộ phận, như thể hiện trong hình 1 tôi. 1, sau đó diện tích dưới đường cong (tích xấp xỉ) được đưa ra bởi | Problems Differential Equations 253 1. Solve the first-order differential equation y -xy2 with y 0 1 with steps of h 0.2 to estimate the value of y 1 using a Euler s method and b the Runge-Kutta method. 2. In this chapter we have examined two methods of solving differential equations. There are many more. One of these is the midpoint method which uses T l y hf xn 2h yn Vihf xn ynỴ Develop a worksheet to use this method to solve y xy at X 0.5 when the initial condition isy 0 I We solved the same problem in Exercises 1 and 2. Which of the three methods gives the more accurate result with the same h increment 3. An analysis of a certain electric circuit shows it obeys the equations dlị dt 12 10 2 10 dỳdt 10 -20 2 where 0 2 0 0. Use the Runge-Kutta method to approximate and 2 at t 0 to 0.5 seconds in 0.1 second intervals. 4. A rocket has a mass of 2000 kg of which 1500 kg is fuel. It bums the fuel at the rate of 25 kg s and develops a thrust of 5000 N. Construct a worksheet to find how the velocity varies with time. What will be the height when the fuel is all consumed 13 Modelling II Concepts In this chapter we use what we have learnt in the last three chapters to model some practical problems. Exercise 1 The Four-bar Crank Using Solver In this exercise we examine an engineering mechanism used to generate a complex motion from a simple rotational motion. Description The four-bar mechanism see Figure 13.1 consists of three movable links a b and c and a fixed link d. The link a is rotated causing link c to rotate. Objective To find how the output angle 4 depends upon the input angle 0 . Assumptions For the quadrilateral formed by the four links the algebraic sum of the vertical component and the algebraic sum of the horizontal component must equate to zero. This gives the two equations ưsinO ốsinp csinộ o ứcos0 cosp ccosộ í 0 Adding the squares of these gives the Freudenstein equation R2 cos 7 3 cos 0 J
