1. The problem statement, all variables and given/known data 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. 2. Relevant 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. 3. 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!