🚀 go-pugleaf

CORRECTION in (2):

One uses the substitution y = x^(r) to get the solutions to the homogeneous
equation, and then uses a change of INDEPENDENT variable, x = e^(*z), to
obtain the particular solution.

FHBRADLEY & LISBunny wrote in message
<7566F23BB6383151.C17C69ADDD5FC3B1.C49BEF153CBB608A@library-proxy.airnews.ne
t>...
Hi, P,

The answer to your question is that you try and solve problems like the ones
you posed using infinite series methods like:

(1) Let  y =  Sigma[An(x-xo)^(n),n = 0, n = infinity], etc.

(2) Special forms of your equation like Euler's Equation:

(x^2)y''(x) + (bx)y'(x) + cy(x) = g(x)

One uses the substitution y = x^(r) to get the solutions to the homogeneous
equation, and then uses a change of dependent variable, x = e^(*z), to
obtain the particular solution.

(3) Near a Regular Singular Point Part I, one uses the infinite series:

y = Sigma[An(x-xo)^(n+r), n=0, n= infinity]

(4) Near a Regular Singular Part II and Part III -- very complex to
implement so I will leave it out unless there is mucho interest here!

(5) Operator Methods

(6) Special case and Special Substitution methods

FHBRADLEY


posenj@l(ancet.co.za wrote in message <354b90c1.1170824@news.saix.net>...
Hi.

This isn't covered in my math syllabus at university. We've used
D-operators and other methods to solve the following type of d.e.:

a.y''(t) + b.y'(t) + c.y(t) = s(t)

where a, b, c are constant.

But how do you solve

f(t).y''(t) + g(t).y'(t) + h(t).y(t) = s(t)

where f, g, h are not constant. I'm interested mainly in numerical
approximation.

I've tried substituting u(t) = y'(t) and then I get two equations:

1) du = (s/f).dt - (g/f).u.dt - (h/f).y.dt
2) u = dy/dt

And, although it look slike I'm getting somewhere, I'm not sure where
to go from there.

Any ideas?

Thanks,
Kevin Posen
South Africa