Explanation Given: The given initial value problem is y ″ − 9 y = g ( t ) ; y ( 0 ) = 2 , y ′ ( 0 ) = 0 Approach: 1) The transfer function of a linear system is defined as the ratio of the Laplace transform of the output function y ( t ) to the Laplace transform of the input function g ( t ) , under the assumption that all initial conditions are zero. That is, H ( s ) = Y ( s ) / G ( s ) . If the linear system is governed by the differential equation a y ″ + b y ′ + c y = g ( t ) , t > 0 Where a , b , and c are constants. Applying the Laplace transform and then assuming all initial conditions are zero, the equation reduces to ( a s 2 + b s + c ) ( Y ( s ) ) = G ( s ) . Thus, the transfer function for the differential equation is H ( s ) = Y ( s ) G ( s ) = 1 a s 2 + b s + c 2) The impulse response function is given by h ( t ) = L − 1 { H } ( t ) 3) The formula for the solution to the given initial value problem is given by, ( h ∗ g ) ( t ) + y k ( t ) . 4) L { e a t } = 1 s − a Calculation: Consider the initial value problem is, y ″ − 9 y = g ( t ) ; y ( 0 ) = 2 , y ′ ( 0 ) = 0 Compare the above differential equation with the equation a y ″ + b y ′ + c y = g ( t ) . So, this gives a = 1 , b = 0 , c = − 9 Thus, the transfer function of the system is H ( s ) = 1 a s 2 + b s + c = 1 ( 1 ) s 2 + 0 ( s ) − 9 = 1 s 2 − 9 Therefore the transfer functions is H ( s ) = 1 s 2 + 9 . And the impulse response function is h ( t ) = L − 1 { H } ( t ) = L − 1 { 1 s 2 − 9 } ( t ) = L − 1 { 1 ( s + 3 ) ( s − 3 ) } ( t ) = 1 6 L − 1 { ( s + 3 ) − ( s − 3 ) ( s + 3 ) ( s − 3 ) } ( t ) Further simplify the above equation as, h ( t ) = 1 6 L − 1 { ( s + 3 ) − ( s − 3 ) ( s + 3 ) ( s − 3 ) } ( t ) = 1 6 L − 1 { ( s + 3 ) ( s + 3 ) ( s − 3 ) } ( t ) − 1 6 L − 1 { ( s − 3 ) ( s + 3 ) ( s − 3 ) } ( t ) = 1 6 L − 1 { 1 ( s − 3 ) } ( t ) − 1 6 L − 1 { 1 ( s + 3 ) } ( t ) = 1 6 e 3 t − 1 6 e − 3 t Since, 1 2 ( e x − e − x ) = sinh ( x ) So,

Fundamentals of Differential Equations [With CDROM] - 7th Edition

7th Edition

ISBN: 9780321410481

Author: Saff, Edward B., Snider, Arthur David, Nagle, R. Kent

Publisher: Addison Wesley

See similar textbooks

Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

Videos

Question

Chapter 7.8, Problem 24E

To determine

To find:

The transfer function H(s) for the given system and the impulse response function h(t) and the formula for the solution to the given initial value problem.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 7.8, Problem 23E

Chapter 7.8, Problem 25E

Students have asked these similar questions

2.2, 13.2-13.3) question: 5 point(s) possible ubmit test The accompanying table contains the data for the amounts (in oz) in cans of a certain soda. The cans are labeled to indicate that the contents are 20 oz of soda. Use the sign test and 0.05 significance level to test the claim that cans of this soda are filled so that the median amount is 20 oz. If the median is not 20 oz, are consumers being cheated? Click the icon to view the data. What are the null and alternative hypotheses? OA. Ho: Medi More Info H₁: Medi OC. Ho: Medi H₁: Medi Volume (in ounces) 20.3 20.1 20.4 Find the test stat 20.1 20.5 20.1 20.1 19.9 20.1 Test statistic = 20.2 20.3 20.3 20.1 20.4 20.5 Find the P-value 19.7 20.2 20.4 20.1 20.2 20.2 P-value= (R 19.9 20.1 20.5 20.4 20.1 20.4 Determine the p 20.1 20.3 20.4 20.2 20.3 20.4 Since the P-valu 19.9 20.2 19.9 Print Done 20 oz 20 oz 20 oz 20 oz ce that the consumers are being cheated.

T Teenage obesity (O), and weekly fast-food meals (F), among some selected Mississippi teenagers are: Name Obesity (lbs) # of Fast-foods per week Josh 185 10 Karl 172 8 Terry 168 9 Kamie Andy 204 154 12 6 (a) Compute the variance of Obesity, s²o, and the variance of fast-food meals, s², of this data. [Must show full work]. (b) Compute the Correlation Coefficient between O and F. [Must show full work]. (c) Find the Coefficient of Determination between O and F. [Must show full work]. (d) Obtain the Regression equation of this data. [Must show full work]. (e) Interpret your answers in (b), (c), and (d). (Full explanations required). Edit View Insert Format Tools Table

H.w WI M Wz dy A Sindax Sind ①dlmax У тах at 0.75m from A W=6KN/M L=2 W2 = 9 KN/m P= 10 KN B

Chapter 7 Solutions

Fundamentals of Differential Equations [With CDROM] - 7th Edition

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 transfer function H ( s ) for the given system and the impulse response function h ( t ) and the formula for the solution to the given initial value problem.

Solution Summary: The author explains the transfer function of a linear system and the impulse response function for the given initial value problem.

Concept explainers

Videos

To find:

The transfer function H(s) for the given system and the impulse response function h(t) and the formula for the solution to the given initial value problem.

Want to see the full answer?

Chapter 7 Solutions

The transfer function H ( s ) for the given system and the impulse response function h ( t ) and the formula for the solution to the given initial value problem.

Solution Summary: The author explains the transfer function of a linear system and the impulse response function for the given initial value problem.

Concept explainers

Videos

To find: The transfer function H(s) for the given system and the impulse response function h(t) and the formula for the solution to the given initial value problem.

Want to see the full answer?

Chapter 7 Solutions

To find:

The transfer function H(s) for the given system and the impulse response function h(t) and the formula for the solution to the given initial value problem.