Concept explainers
(a)
Program Description: Purpose of the problem is to obtain the approximate values of
(a)
Explanation of Solution
Given information:
The differential problem is
The exact solution of the differential equation is
The two step size is
Calculation:
The differential problem can be represented as,
The Euler’s formula for
The Euler’s formula for
Substitute 0 for
Substitute 0 for
Substitute 0 for
Substitute 0 for
Substitute 1 for
Substitute 1 for
Substitute
Substitute
Therefore, the value of
Conclusion:
Thus, the approximate values of
(b)
Program Description: Purpose of the problem is to find the approximate values of
(b)
Explanation of Solution
Given information:
The differential problem is
The exact solution of the differential equation is
The two step size is
Calculation:
The improved Euler’s formula for predicators
The improved Euler’s formula for predicators
Substitute 0 for
Substitute 0 for
Substitute 0 for
Substitute 0 for
The improved Euler’s formula for correctors is shown below.
The improved Euler’s formula for correctors
Substitute 0 for
Substitute 0 for
Substitute 0 for
Substitute 0 for
Therefore, the value of
Conclusion:
Thus, the approximate values of
(c)
Program Description: Purpose of the problem is to find the approximate values of
(c)
Explanation of Solution
Given information:
The differential problem is
The exact solution of the differential equation is
The two step size is
Calculation:
The Runge-Kutta iteration formulas are shown below.
The Runge-Kutta iteration formulas are shown below
The value of
The value of
Substitute 0 for
Substitute 0 for
Substitute 0 for
Substitute 0 for
Substitute 0 for
Substitute 0 for
Substitute 0 for
Substitute 0 for
Substitute
Substitute
The exact solution
The exact solution
It can be observed that the Runge-Kutta method is more closes to the exact solution of the differential equation.
Conclusion:
Thus, the approximate values of
Want to see more full solutions like this?
Chapter 4 Solutions
Differential Equations: Computing and Modeling (5th Edition), Edwards, Penney & Calvis
- a. For the function and point below, find f'(a). b. Determine an equation of the line tangent to the graph of f at (a,f(a)) for the given value of a. f(x) = 2x°, a = 1 %3D ..... a. f'(a) =arrow_forwardSolve the following equations. Be sure to check the potential solution(s) in the original equation, to see whether it (they) are in the domain. (a) log, (r? –x – 2) = 2arrow_forwardSuppose that a parachutist with linear drag (m=50 kg, c=12.5kg/s) jumps from an airplane flying at an altitude of a kilometer with a horizontal velocity of 220 m/s relative to the ground. a) Write a system of four differential equations for x,y,vx=dx/dt and vy=dy/dt. b) If theinitial horizontal position is defined as x=0, use Euler’s methods with t=0.4 s to compute the jumper’s position over the first 40 s. c) Develop plots of y versus t and y versus x. Use the plot to graphically estimate when and where the jumper would hit the ground if the chute failed to open.arrow_forward
- The following is used to model a wave that impacts a concrete wall created by the US Navy speed boat.1. Derive the complete piecewise function of F(t) and F()The concrete wall is 2.8 m long with a cross-section area of 0.05 m2. The force at time equal zero is 200 N. It is also known that the mass is modeled as lumped at the end of 1200 kg and Young’s modulus of 3.6 GPa2. Use *Matlab to simulate and plot the total response of the system at zero initial conditions and t0 = 0.5 sarrow_forwardDiscuss on the need for numerical approximation of solutions to different equations.arrow_forwardProblem 3 In class, we solved for the vorticity distribution for a "real" line vortex diffusing in a viscous fluid. Integrate this vorticity distribution to find the tangential velocity as a function of radius. Plot the velocity distributions for a a line vortex of circulation 0.5 mls in 20 °C air for times of 1, 10, and 100 seconds.arrow_forward
- With the aid of a diagram derive the secant method formular for solving nonlinear equations and use it to find the root of COS x = 0, on [0, t/2] for four iterations. Show working and tabulate your results.arrow_forward2. Heat conduction in a square plate Three sides of a rectangular plate (@ = 5 m, b = 4 m) are kept at a temperature of 0 C and one side is kept at a temperature C, as shown in the figure. Determine and plot the ; temperature distribution T(x, y) in the plate. The temperature distribution, T(x, y) in the plate can be determined by solving the two-dimensional heat equation. For the given boundary conditions T(x, y) can be expressed analytically by a Fourier series (Erwin Kreyszig, Advanced Engineering Mathematics, John Wiley and Sons, 1993):arrow_forwardThe density of a sample of FCC palladium is 11.98 g/cm3, and its lattice parameter is 3.8902 A...Calculate (a) the fraction of the lattice points that contain vacancies: and (b) the total number of vacancies in a cubic centimeter of Pd.arrow_forward
- Problem 1 The position x as a function of time of a particle that moves along a straight line is given by: r(1) = (-3 + 41)c 0. f1 0.1t The velocity v(t) of the particle is determined by the derivative of r(t) with respect to t, and the accelerationa(t) is determined by the derivative ofv(t) with respect to t. Derive the expressions for the velocity and acceleration of the particle, and make plots of the position, velocity, and acceleration as functions of time for0arrow_forwardConsider the function as given. (a) Use a computer algebra system to graph the function and use the graph to approximate the critical numbers visually. (b) Use a computer algebra system to find f′ and approximate the critical numbers. Are the results the same as the visual approximation in part (a)? Explain.arrow_forwardA reservoir discharging water through sluices at a depth hbelow the water surface area Afor various values has given below: hft1011121314( . .)Asqft9501070120013501530If tdenotes time in minutes, the rate of fall of the surface is given by 48dhhAdtEstimate the time taken for the water level to fall from 14 to 10 ft. above the sluices.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
- Database System ConceptsComputer ScienceISBN:9780078022159Author:Abraham Silberschatz Professor, Henry F. Korth, S. SudarshanPublisher:McGraw-Hill EducationStarting Out with Python (4th Edition)Computer ScienceISBN:9780134444321Author:Tony GaddisPublisher:PEARSONDigital Fundamentals (11th Edition)Computer ScienceISBN:9780132737968Author:Thomas L. FloydPublisher:PEARSON
- C How to Program (8th Edition)Computer ScienceISBN:9780133976892Author:Paul J. Deitel, Harvey DeitelPublisher:PEARSONDatabase Systems: Design, Implementation, & Manag...Computer ScienceISBN:9781337627900Author:Carlos Coronel, Steven MorrisPublisher:Cengage LearningProgrammable Logic ControllersComputer ScienceISBN:9780073373843Author:Frank D. PetruzellaPublisher:McGraw-Hill Education