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t S g Call the solution y₁ (t). b. Observe that the only difference between equations (31) and (32) is the constant -b in equation (31). Therefore, it may seem reasonable to assume that the solutions of these two equations also differ only by a constant. Test this assumption by trying to find a constant k such that y = yı(t) + k is a solution of equation (31). c. Compare your solution from part b with the solution given in the text in equation (17). Note: This method can also be used in some cases in which the constant b is replaced by a function g(t). It depends on whether you can guess the general form that the solution is likely to take. This method is described in detail in Section 3.5 in connection with second- order equations. 6. Use the method of Problem 5 to solve the equation 7. The field mouse differential equation dy -ay+b. dt population in Example 1 satisfies the = ODE dy P dt 2 - - - 450. a. Find the time at which the population becomes extinct if p(0) = 850. b. Find the time of extinction if p(0) = Po, where 0 < Po < 900. Nc. Find the initial population po if the population is to become extinct in 1 year. 8. The falling object in Example 2 satisfies the initial value

t S g Call the solution y₁ (t). b. Observe that the only difference between equations (31) and (32) is the constant -b in equation (31). Therefore, it may seem reasonable to assume that the solutions of these two equations also differ only by a constant. Test this assumption by trying to find a constant k such that y = yı(t) + k is a solution of equation (31). c. Compare your solution from part b with the solution given in the text in equation (17). Note: This method can also be used in some cases in which the constant b is replaced by a function g(t). It depends on whether you can guess the general form that the solution is likely to take. This method is described in detail in Section 3.5 in connection with second- order equations. 6. Use the method of Problem 5 to solve the equation 7. The field mouse differential equation dy -ay+b. dt population in Example 1 satisfies the = ODE dy P dt 2 - - - 450. a. Find the time at which the population becomes extinct if p(0) = 850. b. Find the time of extinction if p(0) = Po, where 0 < Po < 900. Nc. Find the initial population po if the population is to become extinct in 1 year. 8. The falling object in Example 2 satisfies the initial value

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...
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how the so a. dy/dt = -y+5, b. dy/dt = -2y +5, c. dy/dt = -2y + 10, 2. Follow the instructions for Problem 1 for the following initial-value problems: G y(0) yo y(0) = yo a. dy/dt = y-5, y(0) = yo G b. dy/dt = 2y-5, c. dy/dt = 2y - 10, y(0) = yo 3. Consider the differential equation dy/dt = -ay+b, where both a and b are positive numbers. a. Find the general solution of the differential equation. G b. Sketch the solution for several different initial conditions. c. Describe how the solutions change under each of the following conditions: i. a increases. y(0) = yo y(0) = yo ii. b increases. iii. Both a and b increase, but the ratio b/a remains the same. 4. Consider the differential equation dy/dt = ay - b. a. Find the equilibrium solution ye. b. Let Y(t) = y - ye; thus Y(t) is the deviation from the equilibrium solution. Find the differential equation satisfied by Y(t). 5. Undetermined Coefficients. Here is an alternative way to solve the equation lok sa dy dt a. Solve the simpler equation dy dt =ay - b. = ay. (31) reason also c to fin equat C. C the t Note: Th constant can gues method i order eq 6. U: (32) 7. T differe a 1 8. prob
Transcribed Image Text:how the so a. dy/dt = -y+5, b. dy/dt = -2y +5, c. dy/dt = -2y + 10, 2. Follow the instructions for Problem 1 for the following initial-value problems: G y(0) yo y(0) = yo a. dy/dt = y-5, y(0) = yo G b. dy/dt = 2y-5, c. dy/dt = 2y - 10, y(0) = yo 3. Consider the differential equation dy/dt = -ay+b, where both a and b are positive numbers. a. Find the general solution of the differential equation. G b. Sketch the solution for several different initial conditions. c. Describe how the solutions change under each of the following conditions: i. a increases. y(0) = yo y(0) = yo ii. b increases. iii. Both a and b increase, but the ratio b/a remains the same. 4. Consider the differential equation dy/dt = ay - b. a. Find the equilibrium solution ye. b. Let Y(t) = y - ye; thus Y(t) is the deviation from the equilibrium solution. Find the differential equation satisfied by Y(t). 5. Undetermined Coefficients. Here is an alternative way to solve the equation lok sa dy dt a. Solve the simpler equation dy dt =ay - b. = ay. (31) reason also c to fin equat C. C the t Note: Th constant can gues method i order eq 6. U: (32) 7. T differe a 1 8. prob
how the so a. dy/dt = -y+5, b. dy/dt = -2y +5, c. dy/dt = -2y + 10, 2. Follow the instructions for Problem 1 for the following initial-value problems: G y(0) yo y(0) = yo a. dy/dt = y-5, y(0) = yo G b. dy/dt = 2y-5, c. dy/dt = 2y - 10, y(0) = yo 3. Consider the differential equation dy/dt = -ay+b, where both a and b are positive numbers. a. Find the general solution of the differential equation. G b. Sketch the solution for several different initial conditions. c. Describe how the solutions change under each of the following conditions: i. a increases. y(0) = yo y(0) = yo ii. b increases. iii. Both a and b increase, but the ratio b/a remains the same. 4. Consider the differential equation dy/dt = ay - b. a. Find the equilibrium solution ye. b. Let Y(t) = y - ye; thus Y(t) is the deviation from the equilibrium solution. Find the differential equation satisfied by Y(t). 5. Undetermined Coefficients. Here is an alternative way to solve the equation lok sa dy dt a. Solve the simpler equation dy dt =ay - b. = ay. (31) reason also c to fin equat C. C the t Note: Th constant can gues method i order eq 6. U: (32) 7. T differe a 1 8. prob
Transcribed Image Text:how the so a. dy/dt = -y+5, b. dy/dt = -2y +5, c. dy/dt = -2y + 10, 2. Follow the instructions for Problem 1 for the following initial-value problems: G y(0) yo y(0) = yo a. dy/dt = y-5, y(0) = yo G b. dy/dt = 2y-5, c. dy/dt = 2y - 10, y(0) = yo 3. Consider the differential equation dy/dt = -ay+b, where both a and b are positive numbers. a. Find the general solution of the differential equation. G b. Sketch the solution for several different initial conditions. c. Describe how the solutions change under each of the following conditions: i. a increases. y(0) = yo y(0) = yo ii. b increases. iii. Both a and b increase, but the ratio b/a remains the same. 4. Consider the differential equation dy/dt = ay - b. a. Find the equilibrium solution ye. b. Let Y(t) = y - ye; thus Y(t) is the deviation from the equilibrium solution. Find the differential equation satisfied by Y(t). 5. Undetermined Coefficients. Here is an alternative way to solve the equation lok sa dy dt a. Solve the simpler equation dy dt =ay - b. = ay. (31) reason also c to fin equat C. C the t Note: Th constant can gues method i order eq 6. U: (32) 7. T differe a 1 8. prob
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