In Exercises
9.
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Differential Equations
- Example: Calculate the value of (aw/@x) at the point (x, y, z) = (2,-1,1) if:- w = x2 + y2 + z? z3 – xy + yz +y3 = 0 Assuming that x and y are the independent variables.arrow_forward1. Is L = {1 − 2x + 3x^2, x − 2x^2, 2 − 5x + 8x^2} linearly independent? If yes, show why. If not, show a dependence relationship. Is L = {1 + 2x + 2x^2, x − 2x^2, −2 − 3x − 5x^2} linearly independent? If yes, show why. If not, show a dependence relationship.arrow_forwardApplying variation of parameters to the DE: y"-y= e" +eª,we get the following system of equations: u e" + u, e- = 0 || u e" – uze- = e® + e=® What are u1 and u2? (u1 and u2 are functions of x.) O u %3D %3D | O u1 = -e"; uz = -e * %3D O u1 = In sin x (e"); u2 = sec x (e) %3D O u1 =+e-2; u2 %3D 1/2arrow_forward
- F (Y = "can be converted into a linear system in the vicinity of each point of .). Every nonlinear system has its equilibrium points. true Errorarrow_forwardConsider the equation system: y2 + w2 − 2xz = 0 x3 + y3 − z3 + w3 = 0 Prove that exists surrounding neighbourhood of point P = (1, −1, 1, 1) in which the solution (x, y, z, w) can be written with x and z as functions of (y, w)arrow_forwardWrite ÿ – 2ij + 5ý + Y = e', as an autonomous system of first-order equations.arrow_forward
- Find the linearization L(x,y) of the function at each point. f(x,y) = x² + y? + 1 а. (2,1) b. (0,2) а. L(x,y) %3 b. L(x,y) =arrow_forwardIn Exercises 11-28, use Separation of Variables to find the generalsolution. y' - ex+y = 0arrow_forwardTwo chemical engineers, A and B, are working independently to develop a model to predict the viscosity of a product (y) from the pH (x1) and the concentration of a certain catalyst (x2). Each engineer has fit the linear model y =A +hr, + B;x; +r The engineers have sent you output summarizing their results: Engineer A Predictor Constant тР Coef SE Coef 199.2 0.5047 394.7 0.000 -1.569 pH Concent. -4.730 0.5857 -8.08 0.000 0.4558 -3.44 0.007 Engineer B Predictor Coef Constant 199.0 0.548 SE Coef T 363.1 0.000 pH Concent. -3.636 1.952 -1.256 1.983 -0.63 0.544 -1.86 0.112 The engineers have also sent you the following scatterplots of pH versus concentration, but forgot to put their names on them. Concentration Concentration Which plot came from which engineer? How do you know? Which engineer's experiment produced the more reliable results? Explain. a. b.arrow_forward
- Find the linearization L(x,y) of the function at each point. f(x,y) = x? +y? +1 а. (3,4) b. (0,2) а. L(x,y) 3D b. L(x,y) =arrow_forwardDefine Equilibrium solutions (For autonomous equations)?arrow_forwardMatch each linear system with one of the phase plane direction fields. (The blue lines are the arrow shafts, and the black dots are the arrow tips.) ? ✓ | 1. z ' = || a' ? 2. ': = ? 3.' = 4. a: = 11 8] -10 3 1 5 -2 1 -5 -13 10] -10 x2 A x2 с x1 (x2 B 2x2/ D Note: To solve this problem, you only need to compute eigenvalues. In fact, it is enough to just compute whether the eigenvalues are real or complex and positive or negative.arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage