1. The problem statement, all variables and given/known data: A function f : R → R is called “even across x∗ ” if f (x∗ − x) = f (x∗ + x) for every x and is called “odd across x∗ ” if f (x∗ − x) = −f (x∗ + x) for every x. Deﬁne f (x) for 0 ≤ x ≤ ℓ by setting f (x) = (x^2) . Extend f to all of R (i.e., deﬁne f (x) for all real x) in such a way that it is odd across x∗ = 0 and even across x∗ = ℓ. 2. Relevant equations: 1. conservation equations (transport): concentration, ﬂux (a) ﬂow (ﬂux = vu ) — ﬂuid, traﬃc, etc. (b) mixing: diﬀusion/dispersion (probability; ﬂux = D ∇u ) reaction/diﬀusion systems 2. mechanics (Newton’s 3rd Law): force, potential energy, momentum (a) wave equation; ICs and BCs (b) beam, plate equations 3. steady state (equilibrium: balance equations) 4. some other examples . . . (e.g., Cauchy-Riemann equations) **Also studying the heat equation/etc** 3. The attempt at a solution: I understand the difference between "even" and "odd". I have created the following: f(ℓ-x)=f(ℓ+x) even at "ℓ" f(0-x)=-f(0-x) OR f(-x)=-f(x) odd at "0" I think I need to use the above IC's to setup the BC's. Once I have the BC's determined I am not sure how to combine them to find the full equation for f (x) or where to start with f (x) = (x^2). Please help!!!