query: Solving the functional equation f(x+y)=f(x)+f(y)+y√f(x)
positive_passages.docid: 153942
positive_passages.text: Solution of functional equation $f(x+y)=f(x)+f(y)+y\sqrt{f(x)}$
negative_passages.docid: 4178
negative_passages.text: Solve $f(x)f(2x^2) = f(2x^3+x)$
negative_passages.docid: 81921
negative_passages.text: If $f(x + y) = f(x) + f(y)$ showing that $f(cx) = cf(x)$ holds for rational $c$
negative_passages.docid: 76458
negative_passages.text: Solve the following functional equation $f(xf(y))+f(yf(x))=2xy$
negative_passages.docid: 109942
negative_passages.text: Find continuous functions such that $f(x+y)+f(x-y)=2[f(x)+f(y)]$
negative_passages.docid: 131586
negative_passages.text: Hard functional equation: $ f \big ( x y + f ( x ) \big) = f \big( f ( x ) f ( y ) \big) + x $
negative_passages.docid: 41226
negative_passages.text: Find $f:f(xf(x)+f(y))=f(x)^2+y$
negative_passages.docid: 81055
negative_passages.text: Functional Equation simple problem
negative_passages.docid: 62111
negative_passages.text: Functional Equation $f(mn)=f(m)f(n)$.
negative_passages.docid: 69707
negative_passages.text: If $f(x + y) \leq yf(x) + f(f(x))$ for all real numbers $x$ and $y$, then prove that $f(x) = 0$ for all $x ≤ 0$.
negative_passages.docid: 69734
negative_passages.text: how to find all functions such that $f\left( x^{2} - y^{2} \right) = ( x - y )( f(x) + f(y) )$
negative_passages.docid: 129599
negative_passages.text: Functional Equation: Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $(x+y)(f(x)-f(y))=(x-y)f(x+y)$
negative_passages.docid: 81720
negative_passages.text: $f(x) + f(x+f(y)) = y + f(f(x) + f(f(y)))$
negative_passages.docid: 19019
negative_passages.text: Find all functions $f:\Bbb{Q}\rightarrow \Bbb{Q}$ such that $f(x)+f(t)=f(y)+f(z)$ for all rational numbers $x<y<z<t$ that form an AP
negative_passages.docid: 35281
negative_passages.text: $1992$ IMO Functional Equation problem
negative_passages.docid: 116270
negative_passages.text: Functional equations leading to sine and cosine
negative_passages.docid: 48652
negative_passages.text: Solve the functional equation $2f(x)=f(ax)$ for some $a$.