Creating symbolic function with certain properties but unspecified functional form?

I've written code to symbolically log-linearize a non-linear model, and would like to specify several symbolic functions generally, without taking a stance on the functional form. In particular, I need f(1)=f'(1)=0 and f''(1)=k (constant). Is there an easy way to specify these properties of a symbolic function without being more specific about the functional form?

Let me know, thanks.

Answer 1

[Walter Roberson]

Sort of. If you can give slightly more information about the form, you can use dsolve()

Within MuPAD, it would look something like this,

dsolve({f(1) = 0, (D(f))(1) = 0, (D[1, 1](f))(1) = k, (D[1, 1, 1](f))(x) = g(x)}, f(x))

The information I added here is D[1, 1, 1](f))(x) = g(x) -- that is, that the third derivative of f(x) is g(x) for some currently unresolved g(x).

I do not have MuPAD to test with, but Maple would resolve this to

f(x) = Int(Int(Int(g(_z1), _z1 = 1 .. _z1), _z1 = 1 .. _z1)+k*_z1, _z1 = 1 .. x)-k*x+k

here, _z1 is a variable that Maple introduced to supply integration limits.

At some later time, if you know (for example) that g(x) = sin(x), you can make a substitution,

subs(g(_z1)=sin(_z1),f(x))

after which in Maple you would convert(%,int) to change the inert Int() into actual int() (MuPAD does not have a convert() function). Maple would then resolve this particular example to

f(x) = -sin(1)-(1/2)*cos(1)+(1/2)*k+sin(1)*x-cos(1)*x+(1/2)*cos(1)*x^2+cos(x)+(1/2)*k*x^2-k*x

[John]

Great, thank you!