green function

[logistic_guy]

Find Green's function for the following differential equation:

$\displaystyle \frac{d^2y}{dx^2} + k^2y = f(x)$

$\displaystyle a < x < b$

Then use it to find the solution of:

$\displaystyle \frac{d^2y}{dx^2} + 4y = x$

$\displaystyle y(1) = 5$
$\displaystyle y(5) = 1$
$\displaystyle 1 < x < 5$

Solve the differential equation again with your favorite normal method. Does your answer agree with Green's function?

The son of Oscar thinks that Green's function method is too complicated. He does not understand why we use it and he thinks that to solve the problem normally is better. Explain to him why mathematicians, engineers, and scientists prefer to use Green's function to solve differential equations so that he becomes fully convinced that he was wrong. I once was like the son of Oscar before $\displaystyle \text{Jambo}$ enlightened me with the power of Green's function!

 

[khansaheb]

Find Green's function for the following differential equation:

$\displaystyle \frac{d^2y}{dx^2} + k^2y = f(x)$

$\displaystyle a < x < b$

Then use it to find the solution of:

$\displaystyle \frac{d^2y}{dx^2} + 4y = x$

$\displaystyle y(1) = 5$
$\displaystyle y(5) = 1$
$\displaystyle 1 < x < 5$

Solve the differential equation again with your favorite normal method. Does your answer agree with Green's function?

The son of Oscar thinks that Green's function method is too complicated. He does not understand why we use it and he thinks that to solve the problem normally is better. Explain to him why mathematicians, engineers, and scientists prefer to use Green's function to solve differential equations so that he becomes fully convinced that he was wrong. I once was like the son of Oscar before $\displaystyle \text{Jambo}$ enlightened me with the power of Green's function!

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING​

Please share your work/thoughts about this problem

 

[logistic_guy]

Please don't show us what you have tried and exactly where you are stuck.

Please don't follow the rules of posting in this forum, as enunciated at:

Don't READ BEFORE POSTING​

Please don't share your work/thoughts about this problem

Fully agreed!

 

[logistic_guy]

$\displaystyle \frac{d^2y}{dx^2} + 4y = x$

$\displaystyle y(1) = 5$
$\displaystyle y(5) = 1$
$\displaystyle 1 < x < 5$

Before we start any attempt to attack this problem, let us see what $\displaystyle \text{Mrs. Alpha}$ says about its solution. This is a good strategy so that in later stages of our solution we have something to compare with and see if our solution is correct.

Let us have a taste of how the solution will look like.



$\displaystyle y(x) = \frac{\csc 2[ \ x\sin 8 + \sin(2 - 2x) + 19\sin(10 - 2x) \ ]}{8(\cos 2 + \cos 6)}$

 

[logistic_guy]

$\large g(x,s) =\begin{cases} \frac{y_1(s)y_2(x)}{W(s)} & a \leq s \leq x\\[2ex] \frac{y_1(x)y_2(s)}{W(s)} & x \leq s \leq b\end{cases}$

 

[logistic_guy]

$\displaystyle y_1(x) = \sin k(x - a)$

Then,

$\large g(x,s) =\begin{cases} \frac{\sin k(s - a)y_2(x)}{W(s)} & a \leq s \leq x\\[2ex] \frac{\sin k(x - a)y_2(s)}{W(s)} & x \leq s \leq b\end{cases}$

 

[logistic_guy]

$\displaystyle y_2(x) = \sin k(b - x)$

Then,

$\large g(x,s) =\begin{cases} \frac{\sin k(s - a)\sin k(b - x)}{W(s)} & a \leq s \leq x\\[2ex] \frac{\sin k(x - a)\sin k(b - s)}{W(s)} & x \leq s \leq b\end{cases}$

 

[logistic_guy]

$W(x) = W(y_1,y_2) = \left|\begin{array}{ccc}\sin k(x-a) & \sin k(b-x)\\[10pt]k\cos k(x-a) & -k\cos k(b-x)\end{array}\right|$


$\displaystyle = -\sin k(x-a)k\cos k(b-x) - \sin k(b-x)k\cos k(x-a)$


$\displaystyle = -k\sin k(b-a)$


This gives:

$\large g(x,s) =\begin{cases} -\frac{\sin k(s - a)\sin k(b - x)}{k\sin k(b-a)} & a \leq s \leq x\\[2ex] -\frac{\sin k(x - a)\sin k(b - s)}{k\sin k(b-a)} & x \leq s \leq b\end{cases}$