3. Find all critical points of dt dt with the constraint PP = 8 0 (c and boundary conditions x(0) - 0, x(1)- 3. Hint: Write the Euler Lagrange equation (there is no dependence on t), and then use the...
(1 point) a. Consider the differential equation: d2y 0.16y-0 dt2 with initial conditions dt (0)-3 y(0)--1 and Find the solution to this initial value problem b. Assume the same second order differential equation as Part a. However, consider it is subject to the following boundary conditions: y(0)-2 and y(3)-7 Find the solution to this boundary value problem. If there is no solution, then write NO SOLUTION. If there are infinitely many solutions, then use C as your arbitrary constant (e.g....
1. (10 points, part I) Consider the following initial boundary value problem lU (la) (1b) (1c) 0L, t> 0 3 cos ( a(x, 0) (a) Classify the partial differential equation (1a) (b) What do the equations (la)-(1c) model? (Hint: Give an interpretation for the PDE, boundary conditions and intial condition.) c) Use the method of separation of variables to separate the above problem into two sub- problems (one that depends on space and the other only on time) (d) What...
Consider the partial differential equation together with the boundary conditions u(0, t) 0 and u(1,t)0 for t20 and the initial condition u(z,0) = z(1-2) for 0 < x < 1. (a) If n is a positive integer, show that the function , sin(x), satisfies the given partial differential equation and boundary conditions. (b) The general solution of the partial differential equation that satisfies the boundary conditions is Write down (but do not evaluate) an integral that can be used to...
help with all except numbers 21-26 16. Solve the differential equation by using the Cauchy-Euler Equation 17. Find the solution to the given Initial Value Problem using Green's Theorem 0,y'(0)s 0 y(0) y" + 6y' + 9y x, 18. Find the solution to the given Boundary Value Problem y" ty-1, y(O)0, y(1) 19. Solve the system of differential equations by systematic elimination. dy dt dt 20. Use any procedure in Chapter 4 to solve the differential equation subjected to the...
solve the PDE +u= at2 on 3 € (0,L), t > 0, with boundary conditions au 2x2 u(0,t) = 0, u(L, t) = 0 au and initial condition u(x,0) = f(x), at (x,0) = g(x) following the steps below. (a) Separate the variables and write differential equations for the functions (x) and h(t); pick the separation constant so that we recover a problem already studied. (b) Find the eigenfunctions and eigenvalues. (c) Write the general solution for this problem. (d)...
Discretization, ODE solving, condition number. Consider the differential equation 5y"(x) - 2y'(x) +10y(x)0 on the interval x E [0,10] with boundary conditions y(0)2 and y (10) 3 we set up a finite difference scheme as follows. Divide [0,10] into N-10 sub-intervals, i.e. {xo, X1, [0,1,. 10. Denote xi Xo + ih (here, h- 1) and yi E y(x). Approximate the derivatives as follows X10- 2h we have the following equations representing the ODE at each point Xi ,i = 1,...
L-8 29 -15 22] 111 4 3 2 1 10. The differential equations of high order: 2 And boundary conditions fo)-0, f' (0)-0, f'(5)-1, g(o)-1.5, g(5)-1 Can be solved using The Shooting-Newton-Raphson and multivariable Runge-Kutta for a value of (y-1.7), re write the system of equations in the canonical form (i.e. as a set of ODES of first order and its boundary conditions). It is not required to solve the equations, just list the system of first order differential equations...
3. Consider the following problem for 0 < x < 1 uzz = f(x) with inhomogeneous boundary conditions u(0) 1, u( 2 (a) Find a Green's function G(x, zo) for this problem, and write down the solution u(z) in terms of G(x, zo) and (x) (b) Solve the problem directly (by integration) in the case when f(x). Show that this gives the same answer as in part (a). 3. Consider the following problem for 0
2. Heat equation Let ult, 2) satisfy the equation 4472(t, 2) +1, 0<r <1, t>0 with initial condition u(0,2) = 0, 0<x<1, and boundary conditions u(t,0) = 0, u(t,1)= 0, t> 0. This equation describes the temperature in a rod. The rod initially has a temperature of 0 (zero degree Celsius), and is then heated at a uniform rate 1. However, its two endpoints are kept at the temperature of 0 at all times. The unknown function u(t, x) describes...
3. Using separation of variables to solve the heat equation, u- kuxx on the interval 0 < x< 1 with boundary conditions ux(0, t) = 0 and ux(1, t) yields the general solution, 1, 0<x < 1/2 0, 1/2 x<1 Determine the coefficients An (n = 0, 1, 2, . . .) when u(x,0) = f(x) = 3. Using separation of variables to solve the heat equation, u- kuxx on the interval 0