query: Solving the functional equation $f(x + y) + g(x-y) = \lambda g(x) f(y)$
positive_passages.docid: 136475
positive_passages.text: d'Alembert-like functional equation: $f(x+y)+g(x-y)=\lambda f(x)g(y)$
negative_passages.docid: 103019
negative_passages.text: Find all functoins $f: \mathbb R \rightarrow \mathbb R$ such that $\forall x,y \in \mathbb R f(xy+f(x))=xf(y)+f(x)$
negative_passages.docid: 121455
negative_passages.text: $f(x)$ if $f(xy)=f(x) +f(y) +\frac{x+y-1}{xy}$
negative_passages.docid: 409
negative_passages.text: functions satisfying $f(x - y) = f(x) f(y) - f(a - x) f(a + y)$ and $f(0)=1$
negative_passages.docid: 145265
negative_passages.text: A non-zero function satisfying $g(x+y) = g(x)g(y)$ must be positive everywhere
negative_passages.docid: 111556
negative_passages.text: Solving functional equation $f(x+y)+f(x-y)=2f(x)\cos y$?
negative_passages.docid: 33315
negative_passages.text: Many other solutions of the Cauchy's Functional Equation
negative_passages.docid: 55286
negative_passages.text: Functional equation: $f: \mathbb{R} \rightarrow \mathbb{R}$, $f((x + 1) f(y)) = y (f(x) + 1)$
negative_passages.docid: 132093
negative_passages.text: Solve $f(f(x))=f(x)$
negative_passages.docid: 15025
negative_passages.text: Solve $xu_x+(x+y)u_y=1$ when $u(1,y)=y$
negative_passages.docid: 88683
negative_passages.text: Functional equation $f\left(\frac{x+y}{2}\right)+f\left(\frac{2xy}{x+y}\right)=f(x)+f(y)$ implies $2f(\sqrt{xy})=f(x)+f(y)$
negative_passages.docid: 35281
negative_passages.text: $1992$ IMO Functional Equation problem
negative_passages.docid: 122329
negative_passages.text: Determine all functions $f(x)$ such that $f(f(x+y))=f(x)+f(y)$
negative_passages.docid: 120170
negative_passages.text: A late-diverging "approximating solution" for a system of functional equations
negative_passages.docid: 116270
negative_passages.text: Functional equations leading to sine and cosine
negative_passages.docid: 128751
negative_passages.text: Solving for $f(2004)$ in a given functional equation
negative_passages.docid: 29647
negative_passages.text: Do there exist functions satisfying $f(x+y)=f(x)+f(y)$ that aren't linear?