2) Show that a Green's function G(x,y) satisfying the problem a2G = 8(x - y), G...
Problem 3 (12 points): Let D be a bounded domain in R" with smooth boundary. Suppose that K(x, y) is a Green's function for the Neumann . For each x E D, the function y H K(x, y) is a smooth harmonic For each x E D, the normal derivative of the function y K(x, y) . For each z e D, the function y K(x,y)-Г(z-y) is smooth near problem. This means the following: function on D(r satisfies (VyK(x, y).v(b))-arefor...
1. Find the particular solution of the differential equation dydx+ycos(x)=2cos(x)dydx+ycos(x)=2cos(x) satisfying the initial condition y(0)=4y(0)=4. 2. Solve the following initial value problem: 8dydt+y=32t8dydt+y=32t with y(0)=6.y(0)=6. (1 point) Find the particular solution of the differential equation dy + y cos(x) = 2 cos(z) satisfying the initial condition y(0) = 4. Answer: y= 2+2e^(-sin(x)) Your answer should be a function of x. (1 point) Solve the following initial value problem: dy ty 8 at +y= 32t with y(0) = 6. (Find y as...
a) Det6 the green's function for y"(x)+ y(x) = f(x) Y(0)=y(1)=0 b) use the result from ( 1) to solve the differential equation y"(x)+y(x)= x Subject to the same boundary condition as in (a)
Using the method of images please help me solve this problem! 1. Using the method of images, find a Green's function for the Laplace operator in the quadrant r > 0, y > 0 which satisfies G(x, xo) on the boundaries 0 and y 0. 1. Using the method of images, find a Green's function for the Laplace operator in the quadrant r > 0, y > 0 which satisfies G(x, xo) on the boundaries 0 and y 0.
QUESTION 8 Write the function in the form y = f(u) and u = g(x). Then find dy/dx as a function of x. 6 y = 8 X O y = 542 10x16 6 u +878 - u; u = x8, causing O y = u8; u = 5x2.-x -f10x -1) Oy= u®; u = 5x2.6 x = 0(532-60) y = 48: u = 5x2.5 - X 5x2 x dx QUESTION 9 Given y = f(u) and u = g(x),...
1) Does the problem = f(x), wr(0) = ur (1) 0 (1) da2 always have a solution? If not, what condition must f(x) satisfy so that a function u(x) satisfying (1) exists? ( Hint: integrate the equation once between 0 and 1.) In this case, is u(x) unique? ( Hint: does the problem with f = 0 have only one solution?)
12. (8 points) A Graph Satisfying First and Second Derivative Conditions On the figure below, sketch the graph of a function y = f(x) that satisfies: • f(-2) = -3, • f is continuous • F"(x) > 0 on (-00, 2). • f is concave up for 1 > 2, and • lim f(1) = -2. • f'(2) does not exist. 00
8. Find a Green's function for Lu u" +4u, 0< x<T, u(0) = u(#) = 0. 9. Find the general solution of ut+ cu f(x,t) 8. Find a Green's function for Lu u" +4u, 0
Find a function y=f(x) satisfying the given differential equation and the prescribed initial condition. 1 dy dx y(7) = -5 1x + 2
please do not do question 1 but add "Assume...." conditions to #1, thank you, upvote for sure Consider the heat equation on "half-line" 0 <<< with prescribed zero temperature at x = 0 U = kuzx (0,t) = 0, ,0) = f(x) [the initial temperature). Find solution of this boundary value problem in the form u(x, t) = G(x – y, t) – 6(x + y,0)) f(y)dy, where G(x, t) = ome. Hint: Extend f(x) as an odd function f(x)...