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Nepal Engineering College Changunarayan, Bhaktapur email: info@nec.edu.np Subject: Numerical Methods Teacher: Hari K. Shrestha Subject Code: MTH 317.3 Tutorial No.: 5 Title: Solution of Ordinary Differential Equations Date: August 10, 2007 One Step methods (Taylor, Picard, Euler, Heun, Runge Kutta) to solve first order initial value problems. 1. Use the Taylor method to solve the equation y’ = x² + y² for x = 0.25 and x = 0.5 given y(0) = 1. 2. Find y(0.1) and y(0.2) using Taylor method to solve the equation y’ = x²y - 1, y(0) = 1. 3. Use Picard’s method to solve the following equations, and find y(0.1), y(0.2) and y(1). a) y’(x) = x² + y², y(0) = 0 b) y’(x) = xey, y(0) = 0 4. Use Euler’s method to solve y’ = 3x² + 1, y(1) = 2. Take step size = 0.5 and estimate y(2). 5. Solve the equation y’ = x + y, y(0) = 1 by Euler’s method. Taking step size = 0.5 and 0.25, find y(1). Compute error in both cases. Compare the results with exact solution (y = 2ex – x – 1). 6. The current I in a circuit having a resistance R and inductance L is given by the differential equation as, dI/dt = (E sin t - RC) / L where E = 100 V, L = 1.5 H, = 500, C = 1 and R = 90 Ohm. Initially, I = 0 at t = 0. By the use of a suitable numerical method, write a flow chart and a computer program in C language to determine current at various times and print. 7. Use Heun’s method to solve y’ = 2y/x, y(1) = 2. Take step size = 0.25 and estimate y(2). 8. A simply supported beam of effective span 6 m is loaded by a system of load q(x) = h x² / l², where h = 10, l = 6 m. Using a suitable numerical method, flowchart and suitable program to determine shear force V at x = 0(1)6. Use differential equation EI (dv / dx) = q(x). 9. Solve the differential equation y’ = x² y + 2x, given y(0) = 0 using Runge Kutta method. 10. Solve the initial value problem y’ = 0.5 (1 + x) y², y(0) = 1, for x = 0(0.1)0.6 using midpoint method formula. 11. Find y(0.1) using Euler’s method when dy/dx = x² + y² and y(0) = 1. Also obtain y(0.1) using Runge-Kutta method using above equation with given initial condition. State the reasons for the difference in the result. 12. Solve the initial value problem y’ = x y + y² given that y(0) = 1 for x = 0(0.1)0.3 using RK4 method. Higher order initial value problems: 13. Solve the following 2nd order differential equation by fourth order Runge-Kutta method. d 2q dq q L 2 R 5 where L=1, C= 0.25, R= 0.5 for t = 0 to 0.2 with a step of 0.1, given q(0) = 0 and q ' (0) = 0 dt dt c 14. Using Runge-Kutta (RK-4) method solve y” = x y’ + y², y(0) = 1, y’(0) = 2, use step size of 0.1 to find y(0.2). 15. Solve the following system of simultaneous equation with associated initial conditions using fourth order Runge-Kutta (RK-4) nethod. y’ = y – z y(0) = 0 z’ = -y + z z (0) = 0 in the range 0 x 1 with step size of 0.5. 16. Solve y” – x² y’ – 2xy = 0; y(0) = 1, y’(0) = 0 for y(0.1) using Runge Kutta (RK4) method. Shooting method to solve Boundary Value Problems (BVP): 17. Using shooting method, solve the equation a) d²y/dx² = 6x, y(1) = 2, y(2) = 9 in the interval (1,2). b) d²y/dx² = 3x + 4y, y(0) = 1, y(1) = 1. Take step size = 0.25 c) d²y/dx² + 2 (dy/dx) – y/2 = 2.5, y(0) = 10, y(10) = 6. d) x² (d²y/dx²) – 2x (dy/dx) + x² sin x = 2y, y(1) = 1, y(2) = 2. Estimate y(1.25), y(1.5) and y(1.75). 18. Using the shooting method, find the solution of the BVP x² y” + x y’ = 1, given y(1) = 0, y(1.4) = 0.0566 2 19. Solve d²y/dx² = e x , y(0) = 0, y(1) = 0 for x = 0.25, 0.5, and 0.75, using the shooting method. d4y 20. The deflection of a beam is governed by the equation 81 y f(x) dx 4 where f(x) is given by the table x 1/3 2/3 1 f(x) 81 162 243 and boundary condition y(0) = y’(0) = y”(1) = y’”(1) = 0. Evaluate the deflection at the pivotal points of the beam using three sub-intervals. 21. Write an algorithm and computer program in any one of the high level programming language to find the solution of second order differential equation y” = f(x,y) by RK-4 method as boundary value problem. Adopt boundary values y(xo) = yo and y’(xo) = y’o. Submission Deadline: August 15, 2007. Attempt only even numbered problems for submission. NM_Tutorial5.doc