Chapter 8: Q 12-12P (page 465)

Question: Find the solution of ( 12.7 ) with $y (0) = y (π / 2) = 0$ when the forcing function is given f(x).
f (x) = sec x.

Short Answer

Expert verified

The value of by forcing the function $f (x) = \sec x$ is $y (x) = \cos (x) \ln (\cos (x)) + \sin (x) (x - π / 2)$ .

Step by step solution

Given information

The given expressions are f (x) = sec x..

Definition of Green Function

In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.

Solve the given function

$f (x) = \sec (x) Hence, we have \begin{matrix} y (x) = - \cos (x) \int_{0}^{x} (\sin (x^{'})) \sec (x^{'}) {dx}^{'} - \sin (x) \int_{x}^{π / 2} (\cos (x^{'})) \sec (x^{'}) {dx}^{'} \\ = - \cos (x) \int_{0}^{x} (\sin (x^{'})) \frac{1}{\cos (x^{'})} {dx}^{'} - \sin (x) \int_{x}^{π / 2} 1 {dx}^{'} \\ = - \cos (x) \int_{0}^{x} \frac{- 1}{\cos (x^{'})} dcos (x^{'}) - \sin (x) \int_{x}^{π / 2} 1 {dx}^{'} \\ = - \cos (x) (- {\ln (\cos (x^{'}))|}_{x}^{0} - \sin (x) ({x^{'}|}_{π / 2}^{x} \\ = - \cos (x) (- \ln (\cos (x)) + \ln (1)) - \sin (x) (π / 2 - x) \\ = - \cos (x) (- \ln (\cos (x))) - \sin (x) (π / 2 - x) \\ = \cos (x) \ln (\cos (x)) + \sin (x) (x - π / 2) \end{matrix}$

Which, the solution to the given differential equation which satisfies the stated boundary conditions.

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Question: Find the solution of ( 12.7 ) with $y (0) = y (π / 2) = 0$ when the forcing function is given f(x).
f (x) = sec x.

Short Answer

Step by step solution

Given information

Definition of Green Function

Solve the given function

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Question: Find the solution of ( 12.7 ) with y(0)=y(π/2)=0when the forcing function is given f(x).f (x) = sec x.

Short Answer

Step by step solution

Given information

Definition of Green Function

Solve the given function

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Medical Physics

Kinematics Physics

Thermodynamics

Oscillations

Electricity

Famous Physicists

Study anywhere. Anytime. Across all devices.

Question: Find the solution of ( 12.7 ) with $y (0) = y (π / 2) = 0$ when the forcing function is given f(x).
f (x) = sec x.