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 the expression above.

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 intial condition ( 0 ) X(0) = 1 -1

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. Hardwoods compete by growing taller than the softwoods can and shading new seedlings. They are also more resistant to disease. Can these two types of trees coexist on one tract of forest land indefinitely, or will one type of tree drive the other to extinction? Model the problem by the system of differential equations: dx1/dt = r1*x1 - a1*(x1)^2 - b1*x1*... click for more

Wave equation using dirchlet boundary conditions

Utt means second derivative with respect to t Uxx means second derivative with respect to x The answer must meet all of the Boundary and other conditions. Solve: Utt = Uxx, 0 < x < pi, t > 0 U(0,t) = 0, U(pi,t) = 0 U(x,0) = 0, Ut(x,0) = 1 for 0 < x < pi The solution should be in an infinite sum.

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 eigenvectors. Also, please check your answer and make sure it is correct. Thank you