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# Numerical Analysis

### Solve a Second Order ODE

Find solution of y'' + 8y' = 896sin(8t) + 640cos(8t) with y(0)= 4 and y'(0)=9

### Carrying capacity of a biologists

Biologists stocked a lake with 400 fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake) to be 7000 . The number of fish doubled in the first year. dP/dt = rP(1- p/K) find an expression for the size of the population after t years by determining constant r.

### Important information about mixture

A tank contains 1320L of pure water.A solution that contains .o1kg of sugar per liter enters a tank at the rate 3L/min The solution is mixed and drains from the tank at the same rate. Solve for function of t So far I have the equation:

### Bisection and newton's method

Q Using the bisection method, find the positive root of 2x(1 + x^2)^-1 = arctan x. Using this root as x0; apply Newton's method to the function f(x) = arctan x: Interpret the results you obtain.

### Minimal Surface Equation

Please see the attached file for the fully formatted problems. Show that Z(x, y) = ln(sin y/sin x) is a solution to the minimal surface equation. (1 + Z)Z1 + 2ZXZZX + (1 + Z)Z = 0, in the region 0 < x < ir, 0 < y < pi. What happens on the boundary of this region? Suppose we consider a constant multiple of Z(x, y) ? is i

### Mathematics: Numerical Analysis Sample Question

PDE- SEND ANSWER AS ATTACHMENT What happens on the boundary of the region? Suppose we consider a constant multiple of Z(x, y). Is it still a solution of the PDE? See attachment for question and details

### Solve the Initial Value Problem

Y'' + 2y' + y = 0 y(0)=1 y'(0)=-3.

### Elliptic Boundary Value Problem: Laplace and Polar Coordinates

(lap) means the Laplacian Vrr means the second derivative of V with respect to r V(theta theta) means the second derivative of V with respect to theta Solve: (lap)V(r,theta)= Vrr+(1/r)Vr+(1/r^2)V(theta theta)=0 0 < r < 1, -(pi) < theta < pi V(1,theta) = {1, -(pi/2) < theta < (pi/2) {0, elsewhere Ple

### Elliptic Boundary Value Problem

Uxx means second derivative with respect to x Uyy means second derivative with respect to y Uxx + Uyy = 0, 0 < x < pi, 0 < y < pi U(x,0) = 0, U(x,pi) = 1, 0 < x < pi U(0,y) = 0, U(pi,y) = 1 0 < y < pi I know the problem has to be broken into 2 separate problems using U = V + W with zero conditions on 3 sides fo

### Solve an IVP ODE using the Method of Variation of Parameters

Please see the attached file for the fully formatted problems. Solve an IVP ODE using the method of variation of parameters Find the solution of the system X' using the method of variation of parameters 2 0 0 cos(t) X' = -1 0 -1 X + sin(t) 1 1 2 e^-t that satisfies the int