Chapter 8: Q 12-10P (page 465)

Question: Solve (12.12)andto get (12.15). Hint: Use Cramer's rule (Chapter 3, Section 3); note that the denominator determinant is the Wronskian [Chapter 3 , equation (8.5)) of the functionssin xand cos x.

Short Answer

Expert verified

The value of $A (x^{'})$ and $B (x^{'})$ from equation $A (x^{'}) \sin x^{'} - B (x^{'}) \cos x^{'} = 0$ and $- A (x^{'}) \cos x^{'} - B (x^{'}) \sin x^{'} = 1$ is $A (x^{'}) = - \cos x^{'}$ is $B (x^{'}) = - \sin x^{'}$ .

Step by step solution

Given information

The given expressions are $A (x^{'}) \sin x^{'} - B (x^{'}) \cos x^{'} = 0 - A (x^{'}) \cos x^{'} - B (x^{'}) \sin x^{'} = 1$ .

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

Let's first rewrite the equations in a way:

$\begin{array}{l} A (x^{'}) \sin x^{'} - B (x^{'}) \cos x^{'} = 0 \\ A (x^{'}) \cos x^{'} + B (x^{'}) \sin x^{'} = - 1 \end{array}$

Now we can use the Cramer's rule:

$\begin{matrix} A (x^{'}) = \frac{|\begin{matrix} 0 & - \cos x^{'} \\ - 1 & \sin x^{'} \end{matrix}|}{|\begin{matrix} \sin x^{'} & - \cos x^{'} \\ \cos x^{'} & \sin x^{'} \end{matrix}|} \\ = \frac{- \cos x^{'}}{\sin^{2} x^{'} + \cos^{2} x^{'}} \\ = - \cos x^{'} \end{matrix}$

$\begin{matrix} B (x^{'}) = \frac{|\begin{matrix} \sin x^{'} & 0 \\ \cos x^{'} & - 1 \end{matrix}|}{|\begin{matrix} \sin x^{'} & - \cos x^{'} \\ \cos x^{'} & \sin x^{'} \end{matrix}|} \\ = - \sin x^{'} \end{matrix}$

Notice that the denominator is the Wronskian of $\sin x^{'}$ and $\cos x^{'}$ because $d (\sin x^{'}) / d x^{'} = \cos x^{'}$ and $d (- \cos x^{'}) / d x^{'} = \sin x^{'}$ .

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Question: Solve (12.12)andto get (12.15). Hint: Use Cramer's rule (Chapter 3, Section 3); note that the denominator determinant is the Wronskian [Chapter 3 , equation (8.5)) of the functionssin xand cos x.

Short Answer

Step by step solution

Given information

Definition of Green Function

Solve the given function

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