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r(1) = 2 y(0) = 1 13. y'=xy³ (1+x²)-1/2,

r(1) = 2 y(0) = 1 13. y'=xy³ (1+x²)-1/2,

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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In each of Problems 1 through 8, solve the given differential equation. x² y y' + y² sinx = 0 y' = cos²(x) cos² (2y) xy' = (1 - y²) 1/2 dy x-e-x dx y + ey x² 1+ y² 1. 2. 3. 4. 5. 6. 7. y' = dy dx dy dx dy dx = = باد 8. In each of Problems 9 through 16: form. a. Find the solution of the given initial value problem in explicit Gb. Plot the graph of the solution. c. Determine (at least approximately) the interval in which the solution is defined. 9. y' = (1-2x) y², 10. y' = (1-2x)/y, 11. xdx+ye-*dy = 0, y(0) = 1 12. dr/d0 = r²/0, r(1) = 2 13. y'=xy³ (1+x²)-1/2, y(0) = 1 14. y' = 2x/(1+2y), y(2) = 0 15. y' = (3x² - e*)/(2y-5), y(0) = 1 16. sin(2x) dx + cos(3y) dy = 0, y(π/2) = π/3 Some of the results requested in Problems 17 through 22 can be obtained either by solving the given equations analytically or by plotting numerically generated approximations to the solutions. Try to form an opinion about the advantages and disadvantages of each approach. G 17. Solve the initial value problem y' y(0) = -1/6 y(1) = -2 = y' = = 1+3x² Зу2 — бу and determine the interval in which the solution is valid. Hint: To find the interval of definition, look for points where the integral curve has a vertical tangent. G 18. Solve the initial value problem y(0) = 1 3x² 3y² - 4' y(1) = 0 and determine the interval in which the solution is valid. Hint: To find the interval of definition, look for points where the integral curve has a vertical tangent. G 19. Solve the initial value problem and determine where the solution attains its minimum value. G 20. Solve the initial value problem 2- et 3+2y y' = 2y² + xy², y(0) = 1 y(0) = 0 and determine where the solution attains its maximum value. G 21. Consider the initial value problem G 22. y' = y' = dQ dt = ty(4- y) 3 a. Determine how the behavior of the solution as t increases depends on the initial value yo. b. Suppose that yo= 0.5. Find the time T at which the solution first reaches the value 3.98. 9 Consider the initial value problem ty(4- y) y' = 1+t a. Determine how the solution behaves as t →∞. b. If yo = 2, find the time T at which the solution first reaches the value 3.99. c. Find the range of initial values for which the solution lies in the interval 3.99 < y < 4.01 by the time t = 2. 23. Solve the equation loa taidanos a y(0) = yo. = ", y(0) = yo > 0. dy ay + b dx cy + d where a, b, c, and d are constants. 24. Use separation of variables to solve the differential equation =r(a+bQ), Q(0) = 20, behaves as t→∞ where a, b, r, and Qo are constants. Determine how the solution dy/dx Homogeneous Equations. If the right-hand side of the equation f(x, y) can be expressed as a function of the ratio y/x only, then the equation is said to be homogeneous.¹ Such equations can always be transformed into separable equations by a change of homogeneous equations. the dependent variable. Problem 25 illustrates how to solve first-order contexts. The homogeneous equations considered here have nothing to do with The word "homogeneous" has different meanings in different mathematical the homogeneous equations that will occur in Chapter 3 and elsewhere.
Transcribed Image Text:In each of Problems 1 through 8, solve the given differential equation. x² y y' + y² sinx = 0 y' = cos²(x) cos² (2y) xy' = (1 - y²) 1/2 dy x-e-x dx y + ey x² 1+ y² 1. 2. 3. 4. 5. 6. 7. y' = dy dx dy dx dy dx = = باد 8. In each of Problems 9 through 16: form. a. Find the solution of the given initial value problem in explicit Gb. Plot the graph of the solution. c. Determine (at least approximately) the interval in which the solution is defined. 9. y' = (1-2x) y², 10. y' = (1-2x)/y, 11. xdx+ye-*dy = 0, y(0) = 1 12. dr/d0 = r²/0, r(1) = 2 13. y'=xy³ (1+x²)-1/2, y(0) = 1 14. y' = 2x/(1+2y), y(2) = 0 15. y' = (3x² - e*)/(2y-5), y(0) = 1 16. sin(2x) dx + cos(3y) dy = 0, y(π/2) = π/3 Some of the results requested in Problems 17 through 22 can be obtained either by solving the given equations analytically or by plotting numerically generated approximations to the solutions. Try to form an opinion about the advantages and disadvantages of each approach. G 17. Solve the initial value problem y' y(0) = -1/6 y(1) = -2 = y' = = 1+3x² Зу2 — бу and determine the interval in which the solution is valid. Hint: To find the interval of definition, look for points where the integral curve has a vertical tangent. G 18. Solve the initial value problem y(0) = 1 3x² 3y² - 4' y(1) = 0 and determine the interval in which the solution is valid. Hint: To find the interval of definition, look for points where the integral curve has a vertical tangent. G 19. Solve the initial value problem and determine where the solution attains its minimum value. G 20. Solve the initial value problem 2- et 3+2y y' = 2y² + xy², y(0) = 1 y(0) = 0 and determine where the solution attains its maximum value. G 21. Consider the initial value problem G 22. y' = y' = dQ dt = ty(4- y) 3 a. Determine how the behavior of the solution as t increases depends on the initial value yo. b. Suppose that yo= 0.5. Find the time T at which the solution first reaches the value 3.98. 9 Consider the initial value problem ty(4- y) y' = 1+t a. Determine how the solution behaves as t →∞. b. If yo = 2, find the time T at which the solution first reaches the value 3.99. c. Find the range of initial values for which the solution lies in the interval 3.99 < y < 4.01 by the time t = 2. 23. Solve the equation loa taidanos a y(0) = yo. = ", y(0) = yo > 0. dy ay + b dx cy + d where a, b, c, and d are constants. 24. Use separation of variables to solve the differential equation =r(a+bQ), Q(0) = 20, behaves as t→∞ where a, b, r, and Qo are constants. Determine how the solution dy/dx Homogeneous Equations. If the right-hand side of the equation f(x, y) can be expressed as a function of the ratio y/x only, then the equation is said to be homogeneous.¹ Such equations can always be transformed into separable equations by a change of homogeneous equations. the dependent variable. Problem 25 illustrates how to solve first-order contexts. The homogeneous equations considered here have nothing to do with The word "homogeneous" has different meanings in different mathematical the homogeneous equations that will occur in Chapter 3 and elsewhere.
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