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In problem 9-14, use Lyapunov’s direct method to determine the stability of the zero solution for the given equation.
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Fundamentals of Differential Equations and Boundary Value Problems
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- 1. Solve the following quadratic equations by factoring. a. x - 9 = 0 e. 4x2 + 9 = 12x %3D b. x? - 4x = 0 f. 3x2 = 13x + 10 c. 3x2 25x g. (x + 1)(x + 3) = -1 %3D d. x2 + x = 20 h. x(x + 2) = 3x(x – 1) – 3arrow_forwardA particular solution of x¨ − 3x˙ + 2x = t2 + 1 is given byarrow_forward1. Solve the equation. If there is no solution or all real numbers are solutions, state that. (a) 4(y + 5) = 21 + 4y %3D (b) 5x + 7 = 2x + 7 (c) 5x – (2x – 8) = 35 (d) 4x + 7 = 7(x + 1) – 3xarrow_forward
- a. Factor f(x) = x + 4x - * - 20x - 20 into factors of the form (x - c), given that -2 is a zero. b. Solve. x* + 4x – x - 20x – 20 = 0arrow_forward(4) Find B such that the equation ßx² + x + B = 0 has a repeated real solution.arrow_forwardOn the basis of data from 1990 to 2006, the median income y in year x for men and women is approximated by the equations given below, where x = 0 corresponds to 1990 and y is in constant 2006 dollars. If these equations remain valid in the future, in what year will the median income of men and women be the same? Men: Women: - 237x + 2y = 63,300 - 850x + 3y = 46,199 The median income of men and women will be the same in the yeararrow_forward
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- No. 25 only Show your step-by-step solution.arrow_forwardSolve for x : 1 4 4 x2 + 5x + 6 9 4x + 8 (no fractions) (in the form ax² + bx +c = 0) (type: "x=? is not applicable" if one of the solutions is not applicable) Warrow_forwardA company finds that the number of new products it develops per year depends on the size of its annual R&D budget, x (in hundreds of thousands of dollars), according to the formula given below. n(x) = -1 + 8x + 4x2 – 0.6x3 (a) Find n"(1) and n"(3). n"(1) n"(3) Interpret the results. With $100,000 invested in R&D, the number of new products per $100,000 is increasing at a rate of new products per $100,000 per $100,000, and with $300,000 invested in R&D, it is decreasing at a rate of new products per $100,000 per $100,000. (b) Find the size of the budget that gives the largest rate of return as measured in new products per dollar (again, called the point of diminishing returns). (Round your answer to the nearest cent.) $arrow_forward
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