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d Q3) For the initial value problem (IVP): y ( (b) = t2, ,...

d Q3) For the initial value problem (IVP): y ( (b) = t2, , y(0) = 1 with dt 0 < t < 1, h = 0.5, exact and the exact solution the solution y (t) = 1/3 1 +1, , 3 evaluate the following (using four-digit rounding arithmetic): (a) with the Taylor's method of order two, approximate ...

Problem 1. Consider the function f: R² R², , f(...

Problem 1. Consider the function f: R² R², , f(x1,x2) = (x3 - 3x1x2, 3x7x2 - x2) (i) Determine the largest subset of points U C R² on which Df (x) is invert- ible. (ii) Provide an argument why around each x # 0 there exists an open disk B = B(i, r) of some radius r > 0 o...

Suppose that temperature distribution of a rod of length 2n ...

Suppose that temperature distribution of a rod of length 2n is given by u =u (x,t), , with initial temperature distribution u (x,0) , -15XSA, and initial end-point values given by u (-xt, 0) =0, u (0,z) = 0. (More generally, the length can be supposed to be 1. As it turns out, 2n naturally fits t...

EXERCISES 11.4.1. Let F : R³ R and f, g, h : R²...

EXERCISES 11.4.1. Let F : R³ R and f, g, h : R² R be C2 functions. If W = F(x, y,Z), where X = f (p, q), y = g (p, q), and Z = h(p, q), find formulas for Wp, Wq, and Wpp. 11.4.6. Let f : R² R be C2 on R² and set u(r, 0) = f (r cos 0, r sin 0). If f satisfies the Laplace...

2. 2+1+2pt+2pt(ext credit)] (a) For the following matrix g...

2. 2+1+2pt+2pt(ext credit)] (a) For the following matrix given by A = 3 1 -2 2 calculate A 1, A 2, A|| as well as the condition numbers for each norm by hand. Is A well or ill conditioned? (b) Recall the formulas from Theorems 2.7 and 2.8. If you assume that taking the ab- solute value an...

Problem 4 Write the following expressions in the indicial n...

Problem 4 Write the following expressions in the indicial notation and subsequently expand them into their component forms. Note: 0 is a scalar, and A is a symmetric second-order tensor. (1) V20 (2) VxAxt Problem 5 If 0 and a are both scalars and a is a vector, prove the following identities u...

Problem 1: Consider the system dx = - x - ß123 dt d...

Problem 1: Consider the system dx = - x - ß123 dt dy = dt k2 where all parameters are positive. Analyze completely this system. You will find 4 different phase portraits based on the parameter values. Problem 2 (Hard): Consider the following system dy dt = with x for Show that the...

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