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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) .
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 .
Consider the initial value problem (IVP): y'(t) = y2 subject to y(0)=1
Approximate y(1) using Euler's method and step sizes of 0.2. Perform these calculations by hand. What is the exact value of y(1)?
Show that the series solution of the initial value problem approaches the steady state solution of the initial value problem as t ? 8.
For 00, and the Robin boundary conditions ux(0,t)-a0u(0,t)=0 and ux (L,t)+ aLu(L,t)=0
Let g(x) = 0.5x + 1.5 and p0 = 4, and consider fixed-point iteration-Show that the fixed point is P=3
Use the false position method to compute Co, C1 , c2, and C3,ex — 2 — x = 0. [ao. bo] = [-2.4, —1.6]
Use Riemann's method to solve the Cauchy problem: u*xx + 4u*xy +3u*yy = 1, u=1 and u*n = square root of 5 times x.
Compute and plot the first 100 points of the Euler method for h=0.1, 0.18, 0.23, 0.25, 0.26, 0.28, and 0.3. Discuss your findings.
Find the conditions on a to ensure that the iteration xn+1=xn-af(xn) will converge linearly (en+1˜ 1/2 en) to a zero of f if started near the zero.
If a sample has a mass of 200 mg find a function describing the mass that remains after t days When will the mass be reduced to 10 mg?
Use a fixed-point interation method to find an approximation to 3v25 that is accurate to within 10-4.
How do I estimate the abosolute and relative maximum and average errors in the total of the above rounded data?
If b2 - 4ac >0, the quadratic equation ax2 + bx +c = zero has two real solutions x1, x2 given by the typical: x1 = (-b + sqrt(b^2-4ac))/ (2a)
Use the method of characteristics to find the solution to (1) with initial condition u(x,0) = f(x).
Explain the following in your own words: (a + b)2 ¹ a2 + b2. Give a numerical example to illustrate this.
Solve the following ?eigenvalue? problem: d2y/dx2+?2y(x)=0 0 < x <1
Explain why you cannot find f(x,y) such that ?f(x,y) = ( x2+3xy2, 2xy+y3+1).
Consider the heat equation ?u/?t= k(?2u/?x2) ,0=x=L t>0 .
Find the solution u(x,t) of the heat equation: ut = 1/2 uxx
The bending moment M at position x m from the end of a simply supported beam of length L m carrying a uniformly distributed load of w Kn m-1 .
Solve the following PDE: du/dt = d^2u / dx^2,u(x, 0) = sin^2(x), u(0, t) = 0, u(Pi, t) = 0,
Find the particular solution of the differential equation , (dy/dx) + ycos(x) = 6cos(x) w/ y(0) = 8
y'''-11 y'' + 18 y' = 0 w/ y(0) = 1, y'(0) = 3, y"(0) = 4