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In Problems 13–32 use variation of parameters to solve the given nonhomogeneous system.
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Chapter 8 Solutions
Bundle: A First Course in Differential Equations with Modeling Applications, Loose-leaf Version, 11th + WebAssign Printed Access Card for Zill's ... Problems, 9th Edition, Single-Term
- 1. The Lotka-Volterra or predator-prey equations dU = aU – UV, dt (1) AP = eyUV – BV. dt (2) have two fixed points (U., V.) = (0,0), (U., V.) = (- :). The trivial fixed point (0,0) is unstable since the prey population grows exponentially if it is initially small. Investigate the stability of the second fixed point (U..V.) = 6:27 PM 3/3/2021 近arrow_forwardProblem 2: Find values of c₁ and c₂ so that y = c₁et + c₂e-t is a solution to the IVP: y=y", y(1) = 0, y' (1) = 1arrow_forward1. Show that y = ze* + e-2* is a solution of y' + 2y = 2e*. 3arrow_forward
- II. Solve the following Bernoulli's equations. 1. y'(t) = 3y – y. Answer. y =esy and y = 0. 1+ce- 2. y' -ty = y, y(2) = -2. Answer. y =x 2-x2 3. xy' + y + xy = 0, y(1) = 2. Answer. y = x(1+2 In x) 4. y' +y = xy', y(0) = -1. Answer. y = - 5. 2 = dy +2x dx Answer. y = +V-1 – 2x + ce2x, 6. y' + xV = 3y. Answer, y = 2 (; + + c*}, and y = 0. Hint. When dividing the equation by Vy. one needs to check if y = 0 is a solution, and indeed it is. 7. y' + y = -xy². Answer. y = " and y = 0. ce-x-1 8. y' + xy = y", y(1) = -! Answer. y = - 1.7. Numerical Solution by Euler's Method 35 9. The equation dy y? + 2x xp y could not be solved in the preceding problem set because it is not homoge- neous. Can you solve it now? Answer. y =tVcezx – 2x – 1. 10. y' = ex +y. Answer. y = te/x² + c. . 11. Solve the Gompertz population model (a and b are positive constants) dx dt = x (a – b ln x), x> 1. Hint. Setting y = In x, obtain y = a – by. Answer. x(t) = q@/bce-br 12. Solve x(y - e) + 2 = 0. Hint. Divide the…arrow_forward3. Suppose ay" + by' + cy = 0 with y(0) = d and y'(0) = k has a general solution y 4e2 - What are the constants a, b, c, d, and k ?arrow_forwardSolve the given differential equationsarrow_forward
- For each dif erential equation in Problems 1–21, find the general solutionby finding the homogeneous solution and a particular solution. Please DO NOT YOU THE PI method where 1/f(r) * x. Dont do that. Instead do this, assume for yp = to something, do the 1 and 2 derivative of it and then plug it in the equation to find the answer.arrow_forwardExample: Discuss the existence and unique solution for the IVP y'= 3y 3, y (0) =0arrow_forwardProblem #1: Solve the following initial value problem. y = -2y₁ - y2 y = бу1 — 7уг y₁(0) = 5, y₂(0) = 3. Enter the functions y₁(x) and y2(x) (in that order) into the answer box below, separated with a comma. Do not include 'y₁(x) =' or 'y₂(x) =' in your answer. Problem #1: Enter your answer as a symbolic function of x, as in these examples Just Save Problem #1 Your Answer: Your Mark: Submit Problem #1 for Grading Attempt #1 Attempt #2 Attempt #3arrow_forward
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