Consider the boundary value problem for the general second-order equation with constant coefficients
y(a)=YA, y(b)=YB
Let the interval a<x<b divided into n subintervals of width h=(b-a)/n.Using central difference approximations
find the lineer system that must be solved to approximate y2,y3,,,yn
Consider the boundary value problem for the general second-order equation with constant coefficients y(a)=YA, y(b)=YB L...
Finite difference methods are also used to approximate the solution to ordinary differential equations. Consider the boundary value problem for the general second-order equation with constant coefficients d2y dy dr2 dr Let the interval a x approximations b be divided inton subintervals of width h -(b- a)/n. Using the central difference find the linear system that must be solved to approximate y2.y3.....yn Finite difference methods are also used to approximate the solution to ordinary differential equations. Consider the boundary value...
1 6. The general form of a linear, homogeneous, second-order equation with constant coefficients is dy dy form. ns (b) Show that if q关0, then the origin is the only equilibrium point of the sys (c) Show, that if q关0, then the only solution of the second-order equation constant is y(t) = 0 for all 1.
Find a second order linear equation L(y) = f(t) with constant coefficients whose general solution is: @ y=Cje24 + C261 + te3t @ (a) The solution contains three parts, so it must come from a nonhomogeneous equation. Using the two terms with undefined constant coefficients, find the characteristic equation for the homogeneous equation. (b) Using the characteristic equation find the homogeneous differential equation. This should be the L(y) we're looking for. (c) Since we have used two terms from the...
7. Consider the boundary value problem for the Laplace equation on the strip u (0, y) u (т, y) = 0, = a. Explain why it makes sense to look for a solution of the form b. Find all solutions of the form u(x, y) -ZYn (v)sinnx satisfying c. Among the solutions you found in part (b) find the unique solution u (x, y)-Yn (y) sin n. the Laplace equation and the boundary conditions. (i.e. find Yn. (3).) that satisfies...
Problem 1. Consider the two data collections (x,..-, X) and (y,. ya). The median difference statistic is the median of the (v-x) differences for the mn possible (x, y.) pairings, i= 1, ..., mandj = 1, . . ., n. Construct two specific data collections (x,. Xm) and (y,. .. V) that demonstrate that the median difference statistic is not equal to the difference in the separate medians for the two collections Problem 2. A Venn diagram is a graphical...
7. Consider the boundary value problem for the Laplace equation on the strip u(0, y) u(n,y)=0, = a. Explain why it makes sense to look for a solution of the form b. Find all solutions of the form u(x,y) = Σ Yn (y) sin nx satisfying c. Among the solutions you found in part (b) find the unique solution u (x, y) = Σ Y, (y) sin na. the Laplace equation and the boundary conditions. (i.e. find Yn (y).) that...
4. Consider the boundary value problem defined by the partial differential equation д?и д?и = 0, ду? y > 0, да? with boundary conditions u(0, y) = u(T,y) = 0, u(x, 0) = 1 and limy-v00 |u(x, y)|< 0o. (a) Use separation of variables to find the eigenvalues and general series solution in terms of the normal modes. (b) Impose the inhomogeneous boundary condition u(x,0) = 1 to find the constants in the general series solution and hence the solution...
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,...
Please provide code and final answer. The code provided solves the boundary value problem 2 dr2 cos(a), J(1) , y(5)2.on the interval Toxksusing a Centred approximation of the derivative term and N= 100 nodes 1 we% Matlab code for the solution of Module 2 3 xright=5; 4 N 100; 5 x-linspace(xleft,xright,N); x x'; %this just turns x into a column vector dx- 7 (xright-xleft)/(N-1); %If theres N nodes, theres N-1 separations . 9 yright 2; 10 here is the matrix...
solve problem #1 depending on the given information Consider the following 1D second order elliptic equation with Dirichlet boundary conditions du(x) (c(x)du ) = f(x) (a $15 b), u(a) = ga, u(b) = gb dr: where u(x) is the unknown function, ga and gb are the Dirichlet boundary values, c(x) is a given coefficient function and f(x) is a given source function. See the theorem 10.1 in the textbook for the existence and uniqueness of the solution. 1.1 Weak Formulation...