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Homework Statement
The problem is actually from chemical kinetics, but my problem is solving the differential equation obtained.
(dx(t))/(dt) = -j*x^2-k*x+k*a ; x is a function of t, and j,k,a are all real positive constants.
Homework Equations
I know this is a Ricatti type equation. But this is from a class for chemists who haven't taken any differential equations classes. So I was trying to solve it by separation of variables. So, the x integral you obtain is
∫ dx/ (-j*x^2-k*x+k*a) with x from 0 to x.
The Attempt at a Solution
I integrate this by means of partial fraction decomposition to obtain the answer in terms of natural log, and in terms of Δ (the discriminant of the second order polynomial)
∫=(j/√Δ)ln[(2jx+k-√Δ)/(2jx+k+√Δ)] + c (this is the indefinite integral)
but the answer expected to obtain is in terms of arctanh (inverse hyperbolic tangent function), I know they're related by arctanhx = ln(1+x/1-x)/2 , but I can't see how to relate my answer to this.
I attach basically the same information I already wrote but in paper, and the expected solution for the differential equation, even though I know that if I can relate the ln answer to the arctan one the rest of the problem is just solving for x.
Thanks for the help!