5. For the last problem, I want to show you how you can deal with non-homogeneou

Question

5. For the last problem, I want to show you how you can deal with non-homogeneous boundary conditions. We'll just do an example but the principle is the same. Suppose that you want to solve the problem y" + 2y = 0, y(0) = a, y(1) = b for a and b nonzero constants. a) First find a function that just satisfies the boundary conditions. For this example, find a linear function l(x) that satisfies 1(0-a and 1(1) = b. b) Now write the solution of (in the form y(x) ()Y(). Plug this into () and derive a new differential equation for Y(x) and show that it has homogenous boundary conditions: y(0)-0 and Y(1) = 0 c) Use eigenfunction expansion to solve for Y(x

Explanation / Answer

a) The linear function will be of the form

l(x) = px + q

l(0) = p(0) + q = a, hence value of q is equal to a

l(1) = p(1) + q = b, hence the value of p will be equal to (b-a)

Therefore, the l(x) linear function will be

l(x) = (b-a)x + a

b)

Assuming the solution is of the form

y(x) = l(x) + Y(x), then it must satisfy the differential equation

y'(x) = l'(x) + Y'(x), similarly y''(x) = l''(x) + Y(x)

Since l(x) is a linear function, hence l''(x) = 0, so y''(x) = Y''(x)

Y''(x) + 2(l(x) + Y(x)) = 0

Now since y(0) = l(0) + Y(0)

a = a + Y(0), so Y(0) = 0

Since y(1) = l(1) + Y(1)

b = (b-a)1 + a + Y(1)

hence Y(1) = 0

Therefore the Y(x) will have homogeneous boundary conditions

c) Y'' + 2Y = 0

writing the characteristic equation

p^2 + 2 = 0

Y(x) = c1*cos(sqrt 2x) + c2 * sin(sqrt 2x)

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