Explanation Procedure used: General solution of a differential equation by undermined coefficients: 1. Consider the given differential equation a 1 y ( n ) + ... + a n − 1 y ′ + a n y = g ( x ) and determine the homogeneous differential equation associated with the DE. 2. Calculate the roots of auxiliary equation of homogeneous equation. 3. Determine g ( x ) . 4. Assume the particular solution to the differential equation in form of an equation. 5. Substitute, the particular solution into the differential equation. 6. Compare the coefficients and determine the general solution. 7. Apply the initial values and determine the unknown coefficients. 8. Substitute, the values of coefficients of complimentary solution into the general solution to find the solution of differential equation with initial values. Calculation: Consider differential equation, y ″ + 4 y = − 2 and determine the associate homogenous equation of differential equation y ″ + 4 y = − 2 that is y ″ + 4 y = 0 . The auxiliary equation corresponding to the homogenous equation y ″ + 4 y = 0 is m 2 + 4 = 0 . Determine the roots of the auxiliary equation m 2 + 4 = 0 , m 2 + 4 = 0 m 2 = − 4 m = ± 2 i Therefore, the roots of the auxiliary equation m 2 + 4 = 0 are m 1 = 2 i and m 2 = − 2 i . Therefore, the complementary function is y c = c 1 cos 2 x + c 2 sin 2 x . Compare the differential equation y ″ + 4 y = − 2 to y ″ + 4 y = g ( x ) . Hence, g ( x ) = − 2 . Assume that a particular solution of the given differential equation is also in the form of a constant function, y p = A y ′ p = 0 , y ″ p = 0 Substitute these values into the differential equation, y ″ + 4 y = − 2 . Therefore, y p ′ ′ + 4 y p = − 2 0 + 4 ⋅ A = − 2 4 ⋅ A = − 2 A = − 2 4 Therefore, a particular solution of differential equation y p = − 1 2 Since, the general solution is given by y = y c + y p . Therefore, y = c 1 cos 2 x + c 2 sin 2 x − 1 2 Hence, the general solution of the differential equation y ″ + 4 y = − 2 is y = c 1 cos 2 x + c 2 sin 2 x − 1 2 . Apply initial condition on general solution of the differential equation, y = c 1 cos 2 x + c 2 sin 2 x − 1 2 y ( π 8 ) = 1 2 c 1 cos 2 ( π 8 ) + c 2 sin 2 ( π 8 ) − 1 2 = y ( π 8 ) c 1 cos ( π 4 ) + c 2 sin ( π 4 ) = 1 2 + 1 2 Further, c 1 2 + c 2 2 = 1 c 1 + c 2 = 2 Differentiate, y

Differential Equations with Boundary-Value Problems

9th Edition

ISBN: 9781337632515

Author: Dennis G. Zill

Publisher: Cengage Learning US

See similar textbooks

Videos

Question

Chapter 4.4, Problem 27E

To determine

To solve: The given initial value problem.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 4.4, Problem 26E

Chapter 4.4, Problem 28E

Students have asked these similar questions

A horse trainer teaches horses to jump by using two methods of instruction. Horses being taught by method A have a lead horse that accompanies each jump. Horses being taught by method B have no lead horse. The table shows the number of training sessions required before each horse performed the jumps properly. Method A 25 23 39 29 37 20 Method B 41 21 46 42 24 44 Method A 45 35 27 31 34 49 Method B 26 43 47 32 40 Use a rank-sum test with a5% level of significance to test the claim that there is no difference between the training sessions distributions. If the value of the sample test statistic R, the rank-sum, is 150, calculate the P-value. Round your answer to four decimal places.

A data processing company has a training program for new salespeople. After completing the training program, each trainee is ranked by his or her instructor. After a year of sales, the same class of trainees is again ranked by a company supervisor according to net value of the contracts they have acquired for the company. The results for a random sample of 11 salespeople trained in the last year follow, where x is rank in training class and y is rank in sales after 1 year. Lower ranks mean higher standing in class and higher net sales. Person 1 2 3 4 5 6 x rank 8 11 2 4 5 3 y rank 7 10 1 3 2 4 Person 7 8 9 10 11 x rank 7 9 10 1 6 y rank 8 11 9 6 5 Using a 1% level of significance, test the claim that the relation between x and y is monotone (either increasing or decreasing). Verify that the Spearman rank correlation coefficient . This implies that the P-value lies between 0.002 and 0.01. State…

Sand and clay studies were conducted at a site in California. Twelve consecutive depths, each about 15 cm deep, were studied and the following percentages of sand in the soil were recorded. 34.4 27.1 30.8 28.0 32.2 27.6 32.8 25.2 31.4 33.5 24.7 28.4 Converting this sequence of numbers to a sequence of symbols A and B, where A indicates a value above the median and B denotes a value below the median gives ABABABABAABB. Test the sequence for randomness about the median with a 5% level of significance. Verify that the number of runs is 10. What is the upper critical value c2?

Chapter 4 Solutions

Differential Equations with Boundary-Value Problems

Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - Given that y=c1+c2x2 is a two-parameter family of...Ch. 4.1 - Find two members of the family of solutions in...Ch. 4.1 - Prob. 7E Ch. 4.1 - Use the general solution of x + 2x = 0 given in...Ch. 4.1 - In Problems 9 and 10 find an interval centered...Ch. 4.1 - In Problems 9 and 10 find an interval centered...

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

The given initial value problem.

Solution Summary: The author explains the general solution of a differential equation by undermined coefficients.

Videos

To solve: The given initial value problem.

Want to see the full answer?

Chapter 4 Solutions