proportionality with more than one variable?

[kochibacha]

if x is direct, indirect or exponentially propotional to A and as well as B

can we write x=kAB ? if we write the equation seperately, we have x=k1A, x=k2B when combined, x²=(k1k2)^1/2 (AB)² then x=k3(AB)^1/2

to see the real complicate example

EX.1 trypsinogen is converted to trypsin in the body where trypsin itself catalyzes its own reaction, assuming it's first order chemical reaction.

let f(t) = amount of trypsin at time t
let F(t) = amount of trypsinogen at time t

write differential equation satisfied by f(t)

in short, f(t) is direct proportional to product itself f(t) and the substrate F(t)

should i write f'(t)=k f(t)F(t) , =k ( f(t)+F(t) ) , = k (f(t)F(t))^1/2 or something else?

 

[JohnZ]

EX.1 trypsinogen is converted to trypsin in the body where trypsin itself catalyzes its own reaction

from the description of the example, conversion of trypsinogen only depends on trypsin. if this is the case, you can probably use the more commonly used model: f'(t) = k * f(t)

 

[kochibacha]

from the description of the example, conversion of trypsinogen only depends on trypsin. if this is the case, you can probably use the more commonly used model: f'(t) = k * f(t)

f'(t) = k * f(t) wont work because amount of trypsin generated depends on trypsin itself and the another function F(t)(trypsinogen which is the progenitor of trypsin). In fact, we can write more generally f'(t) = k*Z(f(t),F(t))

to solve this problem you must concern the context of this problem, let M= F(t)+f(t) the equation becomes y'= k(M-y) ( in this case i assume Z=f(t)*F(t) ) then solving or analyzing this equation is very easy but that's not the point

My point is to discuss what function Z would be. Actually, i think its a simple question if X proportionally depends on A and B how can we express X,A,B in equation forms. Of course, if X proportionally depends on A we can write X=kA but what if A,B or even C,D,E are contributed to X as well can we write X=KA*B*C*D or X=K(A+B+C+D) or X=K(ABC)^1/3. All of these seem to make sense with " X is directly proportional to A and B"