Differential Equations

4. Use Gauss-Jordan elimination to transform the augmented matrix of the following system to RREF. Then use the Nonhomoge- neous Principle to write the solution of the linear system in the form X = X₁ + Xp- vrie 2x1 22 23 2 4x1 + x2 + 5x3 = 4 x1 + x2 + 2x3 = 1 Bo

1. Use the inverse matrix to solve the linear following linear systems. Look carefully at the linear systems to significantly reduce the workload. (a) 2x + 3y + z = 4 3x + 3y + 2 = 8 2x + 4y + z = 5 (b) 2x + 3y+z=0 3x + 3y + z = 0 2x + 4y+z=0

2. Two chemicals are mixed together and as they react the temperature of the mixture is measured. One, two and three hours after the mixing takes place, the temperatures of the chemicals are, respectively, 156.8, 161.7 and 177.2 degrees. (a) Create a system of linear equations for the data to fit the curve T(t) = at² + bt+c, where t is the time in hours and T' is the temperature. You essentially need to find a parabola that passes through the three points (1, 156.8), (2, 161.7), (3, 177.2). (b) Write the system in the form AX = b. (c) Use Cramer's Rule to solve the system. Be sure to at least show the set up for using Cramer's Rule. You can use a computer to compute the determinants if so desired. (d) Write the equation of the parabola. er

S={-2t+t²,8 +1³, -1² +13³, -4+1²} spans Ps. 4. Determine whether the set.

. Determine the solution of equa- tion (4) satisfying the initial conditions y(0) = 1 and y' (0) = 3.

Write the second order inhomogeneous linear differential equation

Determine the nature and location of the stationary point of the function y = x³ +6x² +12x-2. The coordinates of the stationary point are (x, y) =

Q.6) (a) Suppose a region R, that is above the the x-axis, has an area given by A. If that region is rotated about the x-axis, the volume of revolution is given by V₁. If the same region is rotated about the line y = −k (where k > 0), the volume of revolution is given by V₂. Derive an expression for V₂ in terms of V₁, k and A. You should include a suitable diagram in your derivation. Comment on how you would verify that the result you obtained is correct. (b) Consider a function of two variables f(x, y) which has partial derivatives of all order. For all values of t, u and u, the function satisfies f(tu, tv) = t²f(u, v). Derive a simplified expression forin terms of f(x, y). (c) Consider the function f(x) = x and assume that f(0) = 1. Using an appropriate established MacLaurin series, and term by term integration, derive an infinite series expression for the integral cl

Find the equation of the tangent drawn to the graph of f(x)=sin(ln(x)) +5 at z = 6. The equation is of the form y = ax + b. Find a and b to two decimal places:

Find the first and second derivatives of y = cos(4x) + sin(9x). Use the buttons above the response area to be sure you are entering sin and cos in the correct format.