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II. Solve the following Bernoulli's equations. 1. y'(1) = 3y - y. Answer. y= I+cr and y= 0. 2. y' -y = y*, y(2) =-2. 2x Answer. y= 2-x 3. xy' + y+ xy = 0, y(1) = 2. Answer. y= X(1+2 in x) 4. y' + y = xy', y(0) = -1. Answer. y=- dy - +2x %3D dx y Answer. y =tV-1– 2x + ce2x, 6. y' +xVỹ = 3y. Answer. y= - (;+*, and y = 0. +( + + ce«)°. Hint. When dividing the equation by Vy, one needs to check if y - 0 is a solution, and indeed it is. 7. y' +y = -xy, Answer. y= - ce-x-1 ,and y 0. 8. y' + xy = y', y(1) = -! Answer. y= - 1.7. Numerical Solution by Euler's Method 9. The equation 35 dy y + 2x dx could not be solved in the preceding problem set because it is not homoge- neous. Can you solve it now? Answer. y =tVcex – 2x – 1. 10. y' -ex + y. Answer. y= te'Vx + c. . 11. Solve the Gompertz population model (a and b are positive constants) dx =x (a - bln x), x>1. Hint. Setting y = In x, obtain y' = a - by. Answer. x(t) qa/bce -. 12. Solve x(y - e) + 2 = 0. Hint. Divide the equation by e" and then set u = e"Y, obtaining a linear equa- tion for v = v(x). Answer. y=- In (x + ex)

II. Solve the following Bernoulli's equations. 1. y'(1) = 3y - y. Answer. y= I+cr and y= 0. 2. y' -y = y*, y(2) =-2. 2x Answer. y= 2-x 3. xy' + y+ xy = 0, y(1) = 2. Answer. y= X(1+2 in x) 4. y' + y = xy', y(0) = -1. Answer. y=- dy - +2x %3D dx y Answer. y =tV-1– 2x + ce2x, 6. y' +xVỹ = 3y. Answer. y= - (;+*, and y = 0. +( + + ce«)°. Hint. When dividing the equation by Vy, one needs to check if y - 0 is a solution, and indeed it is. 7. y' +y = -xy, Answer. y= - ce-x-1 ,and y 0. 8. y' + xy = y', y(1) = -! Answer. y= - 1.7. Numerical Solution by Euler's Method 9. The equation 35 dy y + 2x dx could not be solved in the preceding problem set because it is not homoge- neous. Can you solve it now? Answer. y =tVcex – 2x – 1. 10. y' -ex + y. Answer. y= te'Vx + c. . 11. Solve the Gompertz population model (a and b are positive constants) dx =x (a - bln x), x>1. Hint. Setting y = In x, obtain y' = a - by. Answer. x(t) qa/bce -. 12. Solve x(y - e) + 2 = 0. Hint. Divide the equation by e" and then set u = e"Y, obtaining a linear equa- tion for v = v(x). Answer. y=- In (x + ex)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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 equation by e' and then set u = e=Y, obtaining a linear equa- tion for v = v(x). Answer. y = - In (x + cx²).
Transcribed Image Text: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 equation by e' and then set u = e=Y, obtaining a linear equa- tion for v = v(x). Answer. y = - In (x + cx²).
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