Mathematics Homework Solutions

Linearization of a function. Attachments in Word.

The distance l from a point at a height h above the Earth's surface to the horizon can be approximated using Pythagoras' theorem by the expression: (Please see the attachment below) (a) Find an expression which serves as a linear approximation for l at h=1000 m. (b) Give two assumptions you think have been made in deriving ...continues

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 ...continues

Modelling Systems : Competing Species

In an unmanaged tract of forest area, hardwood and softwood trees compete for the available land and water. The more desirable hardwood trees grow more slowly, but are more durable and produce more valuable timber. Softwood trees compete with the hardwoods by growing rapidly and consuming the available water and soil nutrients ...continues

Use D'Alembert's Formula

Utt means the second derivative with respect to t Uxx means the second derivative with respect to x Utt = 4Uxx, -(inf) < x < (inf), t > 0 U(x,0) = x, Ut(x,0) = xe^(-x^2) for -(inf) < x < (inf) Please use D'Alembert's Formula and show all work. If there is Fourier series, please show how you got eigenvalues and eige ...continues

Wave equation with mixed boundary conditions

Uxx means second derivative with respect to x Uyy means second derivative with respect to y Uxx + Uyy = 0, 0 < x < pi, 0 < y < 1 Ux(0,y) = 0 = U(pi,y), 0 < y < 1 U(x,0) = 1, U(x,1) = 0, 0 < x < pi Please show all work including how eigenvalues and eigenvectors are derived. Thank you

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 ...continues

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 ...continues

It is an explanation for solving homogeneous linear partial differential equation. Find the General solutions of the equations r = a2t and (D + D’)z = sin x.

Linear Partial Differential Equation (I) Linear Homogeneous Partial Differential Equation with Constant Coefficients Problem 1: Find the General solution of the equation r = a2t. ...continues

It is an explanation for solving Non- Homogeneous Linear Partial Differential Equation with Constant Coefficients. Find the solution of the equation (D2 – D’2 + D – D’)z = e^(2x + 2y).

Linear Partial Differential Equation (II) Non- Homogeneous Linear Partial Differential Equation with Constant Coefficients Problem: Find the solution of the equation (D2 – D’2 + D – ...continues

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