Question: A mammalogist models the population of a species using the functional equation $ h(x + y) = h(x) + h(y) + xy $ for all real $ x, y $. Find all functions $ h: \mathbb{R} o \mathbb{R} $.

["Understanding the Functional Equation: Finding All Mammalogists’ Population Models", "In studying species dynamics, mammalogists often rely on mathematical models to predict population trends. One intriguing functional equation—$ h(x + y) = h(x) + h(y) + xy $—arises when analyzing growth patterns influenced by both linear and interacting terms. This article explores all real-valued functions $ h: \mathbb{R} \ o \mathbb{R} $ satisfying this equation, revealing deep insights into biological modeling through mathematics.", "We begin with the functional equation:", "$$\nh(x + y) = h(x) + h(y) + xy \quad \ ext{for all } x, y \in \mathbb{R}.\n$$", "This equation combines features of Cauchy’s additive functional equation and quadratic correction. Our goal is to find all continuous (and later, all measurable or bounded-in moderate regions) functions satisfying this relation.", "---", "### Step 1: Transform the Equation to a Known Form", "To simplify, define a new function $ f(x) = h(x) + ax^2 + bx $. The quadratic term is motivated by the $ xy $ term on the right-hand side. We aim to eliminate the cross term and reduce the equation to Cauchy’s form.", "But first, fix $ y = 0 $:", "$$\nh(x + 0) = h(x) + h(0) + x \cdot 0 \Rightarrow h(x) = h(x) + h(0) \Rightarrow h(0) = 0.\n$$", "So $ h(0) = 0 $.", "Now use $ y = x $:", "$$\nh(2x) = 2h(x) + x^2.\n$$", "This symmetry suggests a quadratic contribution.", "---", "### Step 2: Guess a Polynomial Form", "Since the equation involves a bilinear term $ xy $, we hypothesize $ h(x) $ is quadratic. Suppose:", "$$\nh(x) = ax^2 + bx.\n$$", "Compute both sides of the original equation:", "Left-hand side:", "$$\nh(x + y) = a(x + y)^2 + b(x + y) = a(x^2 + 2xy + y^2) + b(x + y) = ax^2 + ay^2 + 2axy + bx + by.\n$$", "Right-hand side:", "$$\nh(x) + h(y) + xy = (ax^2 + bx) + (ay^2 + by) + xy = ax^2 + ay^2 + bx + by + xy.\n$$", "Set both sides equal:", "$$\nax^2 + ay^2 + 2axy + bx + by = ax^2 + ay^2 + bx + by + xy.\n$$", "Cancel common terms:", "$$\n2axy = xy \quad \Rightarrow \quad 2a = 1 \Rightarrow a = \frac{1}{2}.\n$$", "Thus, $ h(x) = \frac{1}{2}x^2 + bx $ satisfies the equation for any real $ b $.", "---", "### Step 3: Uniqueness — Show All Solutions Are of This Form", "We now prove that all solutions $ h: \mathbb{R} \ o \mathbb{R} $ are of the form", "$$\nh(x) = \frac{1}{2}x^2 + bx,\n$$", "assuming sufficient regularity (e.g., continuity, or boundedness on an interval, or measurability). Without such assumptions, pathological solutions exist via Hamel bases, but in modeling contexts—especially biological ones like population dynamics—continuous solutions are physically meaningful and expected.", "To prove uniqueness under continuity, suppose $ h $ is continuous.", "From earlier, define $ f(x) = h(x) - \frac{1}{2}x^2 $. Then substitute into the functional equation:", "$$\nh(x + y) = h(x) + h(y) + xy\n\Rightarrow f(x + y) + \frac{1}{2}(x + y)^2 = \left(f(x) + \frac{1}{2}x^2\right) + \left(f(y) + \frac{1}{2}y^2\right) + xy.\n$$", "Expand:", "$$\nf(x + y) + \frac{1}{2}(x^2 + 2xy + y^2) = f(x) + f(y) + \frac{1}{2}x^2 + \frac{1}{2}y^2 + xy.\n$$", "Simplify:", "$$\nf(x + y) + \frac{1}{2}x^2 + xy + \frac{1}{2}y^2 = f(x) + f(y) + \frac{1}{2}x^2 + \frac{1}{2}y^2 + xy.\n$$", "Cancel terms:", "$$\nf(x + y) = f(x) + f(y).\n$$", "So $ f $ satisfies Cauchy’s functional equation. If $ h $ is continuous, then $ f $ is continuous, and the only solutions are linear: $ f(x) = bx $ for some $ b \in \mathbb{R} $.", "Therefore,", "$$\nh(x) = \frac{1}{2}x^2 + bx.\n$$", "---", "### Biological Interpretation", "In mammalogical modeling, $ h(t) $ could represent the population size at time $ t $. The term $ \frac{1}{2}x^2 $ models nonlinear growth influenced by population density interactions, while $ bx $ reflects baseline growth or immigration. The functional equation arises naturally when assuming linear additive change plus a consistent interaction cost proportional to $ xy $, common in adaptive population models.", "---", "### Conclusion", "All real-valued functions $ h: \mathbb{R} \ o \mathbb{R} $ satisfying $ h(x + y) = h(x) + h(y) + xy $ for all real $ x, y $ are quadratic functions of the form:", "$$\nh(x) = \frac{1}{2}x^2 + bx, \quad \ ext{where } b \in \mathbb{R}.\n$$", "This result enables mammalogists and ecologists to precisely characterize population models governed by such relational dynamics, ensuring both mathematical rigor and biological relevance.", "---", "Keywords: functional equation, mammalogist modeling, population dynamics, Cauchy equation, quadratic function, mathematical biology, additive functions\nMeta Description: Find all solutions $ h: \mathbb{R} \ o \mathbb{R} $ to $ h(x + y) = h(x) + h(y) + xy $. The solution is $ h(x) = \frac{1}{2}x^2 + bx $ for constant $ b $."]