differential equation

differential equation use the maple program to slov the equation and make sure that the assignment is chose one type of equation so i chose the differential equation Write a program in Maple ( or Mupad) to use the techniques of Laplace transforms to solve the system of equations d2 x = ax + by dt2 d2 y = cx + dy dt2 for any specified values of a, b, c, d, x(0), y(0), Dx(0), Dy(0). Use examples to test this program and in particular use it in the case a = b = c = d = 1, x(0) = 1, Dx(0) = 0, y(0) = 0, Dy(0) = 1 20 Marks To complete the assignment apply your program to solve ONE of the two physical systems given below. You should choose appropriate and realistic parameters and illustrate your solutions with both an analytical and graphical output. 30 Marks 1. The equations for two identical masses coupled by three springs are d2 x m 2 = -kx - k(x - y) dt d2 y = -k(y - x) - ky dt2 where x and y are the displacements of the masses from their equilibrium position.Initially the masses are displaced from their equilibrium positions and then released so that m x(0) = a, y(0) = b, Dx(0) = Dy(0) = 0 Repeat the calculations with different , small displacements, a, b , different realistic force constants and comment on the output. OR 1 2. The equations for a circuit comprising of two identical LC circuits coupled with a different capacitor ,Cˆ are d2 I1 = -?(1 + a)I1 - ? 2 aI2 dt2 d2 I2 = -?(1 + a)I2 - ? 2 aI1 dt2 where I1 , I2 are the currents in the two circuits and ?= 1 , Lc a= C Cˆ For given, realistic values of L, C, Cˆ solve this system for different initial currents I1 (0), I2 (0) and derivatives I10 (0), I20 (0). The figures below illustrate the physical systems. 2 x y m m k k k md 2 x/dx^2 = -kx - k(x-y), md 2 y/dy^2 = -k(y-x) -ky . Here x and y are the displacements measured from the masses m and k is the common spring constant. L 00000000 C1 L 00000000 C3 C2 d^2x/dt^2= -w^2(1+a)x -w^2a y, d^2y/dt^2 = -w^2(1+a)y - w^ax, a= C2/C1, C1=C3, w^2 = 1/(LC1) Here x an y are the current in the left circuit and right circuit respectively. The C's are the capacitances and L an inductance. 3 Figure 1: Spring and Circuit Differential Equations Simulation of Daily Temperature Variation The internal temperature T (t ) of a building is governed by Newton’s law of cooling and is modelled by the following differential equation: dT ? ?0.2(T ? A(t )), T (0) ? T0 . dt Here t is time (in hours), and A(t ) is the ambient (external) temperature given 2?t by A(t ) ? A0 ? A1 cos . 24 (a) Choose appropriate values for the constants T0 , A0 and A1 . Find the exact solution to the equation (using Maple’s dsolve command - see class notes) and plot it for a time period of three days. [5 marks] (b) Generate approximate solutions to the equation for a step length of 2 hrs using both the Euler method and the Modified Euler method for a period of three days, and plot them against the exact solution. [15 marks] (c) Suppose t ? 0 refers to mid-day on a Monday and a prediction of the temperature is required for 4am the following Thursday morning. Investigate the errors produced by the two numerical methods at this time (compared with the exact solution) when step lengths of 2 hrs, 1 hr and 30 mins are used. Use your values to complete the table below and comment. Steplength h Exact Temp Euler method Temp |Error| Modified Euler method Temp |Error| 2 1 0.5 [10 marks] PLACE THIS ORDER OR A SIMILAR ORDER WITH US TODAY AND GET AN AMAZING DISCOUNT :)