query: Differential calculus: $f(x)+f''(x)=-xg(x)f'(x)$, $\forall x \geq 0$ where $g(x) \geq 0, \forall x \geq 0$.
positive_passages.docid: 151819
positive_passages.text: A twice differentiable function $f$ satisfies $f′′(x)+f(x)=−xg(x)f′(x)$, $\forall x\ge 0$.
negative_passages.docid: 49408
negative_passages.text: Proof of the derivative of ln(x)
negative_passages.docid: 43956
negative_passages.text: Differentiating $f(x)^{g(x)}$
negative_passages.docid: 41226
negative_passages.text: Find $f:f(xf(x)+f(y))=f(x)^2+y$
negative_passages.docid: 112923
negative_passages.text: Let $f:\mathbb{R}\longrightarrow \mathbb{R}$ a differentiable function such that $f'(x)=0$ for all $x\in\mathbb{Q}$
negative_passages.docid: 133682
negative_passages.text: Prove that if $f(t)$ satisfies $f(t)=f'(t)$, then $f(t+s)=f(t)f(s)$ for all $t,s\in R$ without using the fact that $f(t)=c\cdot e^t$.
negative_passages.docid: 39090
negative_passages.text: Finding functions such that $f(g(x)) = f(x)g(x)$
negative_passages.docid: 44060
negative_passages.text: If $\lim_{x \to +\infty} f'(x) = L$, then $\lim_{x \to \infty} \frac {f(x)}{x} = L$
negative_passages.docid: 139216
negative_passages.text: Use the definition of derivative to find $f'(x)$ for $f(x) = x^{1/2}$.
negative_passages.docid: 146480
negative_passages.text: Example of $\deg(fg)<\deg(f)+\deg(g)$
negative_passages.docid: 85739
negative_passages.text: Show that $f(x) = 0$ for all $x \in [a,b]$ given $|f'(x)| \leq C|f(x)| $
negative_passages.docid: 64339
negative_passages.text: Find $f$ where $f'(x) = f(1+x)$
negative_passages.docid: 81720
negative_passages.text: $f(x) + f(x+f(y)) = y + f(f(x) + f(f(y)))$
negative_passages.docid: 49211
negative_passages.text: Solving $f(x+y) = f(x)f(y)f(xy)$
negative_passages.docid: 94944
negative_passages.text: Differentiable functions satisfying $f'(f(x))=f(f'(x))$
negative_passages.docid: 59063
negative_passages.text: $f(f(x))=f(x)$ question
negative_passages.docid: 57026
negative_passages.text: $f(0)=f'(0)=f'(1)=0$ and $f(1)=1$ implies $\max|f''|\geq 4$