Find all twice continuously differentiable functions $f: \mathbb{R} \to (0, \infty)$ satisfying $f''(x)f(x) \ge 2f'(x)^2.$

For the complex-valued function $f(x)$ which is continuous and absolutely integrable on $\mathbb{R}$, define the function $(Sf)(x)$ on $\mathbb{R}$: $(Sf)(x)=\int_{-\infty}^{+\infty}e^{2\pi iux}f(u)du$. (a) Find the expression for $S(\frac{1}{1+x^2})$ and $S(\frac{1}{(1+x^2)^2})$. (b) For any integer $k$, let $f_k(x)=(1+x^2)^{-1-k}$. Assume $k\geq 1$, find constant $c_1$, $c_2$ such that the function $y=(Sf_k)(x)$ satisfies the ODE with second order: $xy''+c_1y'+c_2xy=0$.

$u(t)$ is solution of the following initial value problem. $$\begin{cases} u''(t) + u'(t) = \sin u(t) &\;\;(t>0),\\ u(0)=1,\;\; u'(0)=0 & \end{cases}$$ (1) Show that $u(t)$ and $u'(t)$ are bounded on $t>0$. (2) Find $\lim\limits_{t\to\infty} u(t)$ with proof.

Prove that if the family of integral curves of the differential equation $$ \frac{dy}{dx} +p(x) y= q(x),$$ where $p(x) q(x) \ne 0$, is cut by the line $x=k$ the tangents at the points of intersection are concurrent.

Provided that the roots of the polynom $ X^n+a_1X^{n-1} +a_2X^{n-2} +\cdots +a_{n-1}X +a_n:\in\mathbb{R}[X] , $ of degree $ n\ge 2, $ are all real and pairwise distinct, prove that there exists is a neighbourhood $ \mathcal{V} $ of $ \left( a_1,a_2,\ldots ,a_n \right) $ in $ \mathbb{R}^n $ and $ n $ functions $ x_1,x_2,\ldots ,x_n\in\mathcal{C}^{\infty } \left( \mathcal{V} \right) $ whose values at $ \left( a_1,a_2,\ldots ,a_n \right) $ are roots of the mentioned polynom.

Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function such that $$f(x)=f \left( \frac{x}{2} \right) + \frac{x}{2} f'(x), ~\forall x \in \mathbb{R}.$$ Prove that $f$ is a polynomial function of degree at most one. [hide=Note]The problem was posted quite a few times before: [url]https://artofproblemsolving.com/community/c7h100225p566080[/url] [url]https://artofproblemsolving.com/community/q11h564540p3300032[/url] [url]https://artofproblemsolving.com/community/c7h2605212p22490699[/url] [url]https://artofproblemsolving.com/community/c7h198927p1093788[/url] I'm reposting it just to have a more suitable statement for the [url=https://artofproblemsolving.com/community/c13_contests]Contest Collections[/url]. [/hide]

We want to find functions $p(t)$, $q(t)$, $f(t)$ such that (a) $p$ and $q$ are continuous functions on the open interval $(0,\pi)$. (b) $f$ is an infinitely differentiable nonzero function on the whole real line $(-\infty,\infty)$ such that $f(0)=f'(0)=f''(0)$. (c) $y=\sin t$ and $y=f(t)$ are solutions of the differential equation $y''+p(t)y'+q(t)y=0$ on $(0,\pi)$. Is this possible? Either prove this is not possible, or show this is possible by providing an explicit example of such $f,p,q$.

A pond has been dug at the Indian Statistical Institute as an inverted truncated pyramid with a square base (see figure below). The depth of the pond is 6m. The square at the bottom has side length 2m and the top square has side length 8m. Water is filled in at a rate of $\tfrac{19}{3}$ cubic meters per hour. At what rate is the water level rising exactly $1$ hour after the water started to fill the pond? [img]https://cdn.artofproblemsolving.com/attachments/0/9/ff8cac4bb4596ec6c030813da7e827e9a09dfd.png[/img]

For all real $x$, the real-valued function $y=f(x)$ satisfies \[ y''-2y'+y=2e^x. \] (a) If $f(x)>0$ for all real $x$, must $f'(x) > 0$ for all real $x$? Explain. (b) If $f'(x)>0$ for all real $x$, must $f(x) > 0$ for all real $x$? Explain.

Find an integral formula for the solution of the differential equation $$\delta (\delta-1)(\delta-2) \cdots(\delta -n +1) y= f(x), \;\;\, x\geq 1,$$ for $y$ as a function of $f$ satisfying the initial conditions $y(1)=y'(1)=\ldots= y^{(n-1)}(1)=0$, where $f$ is continuous and $\delta$ is the differential operator $ x \frac{d}{dx}.$

Show that if a connected, nowhere zero sectional curvature of Riemannian manifold, where symmetric (1,1)-tensor of the Levi-Civita connection covariant derivative vanishes, then the tensor is constant times the unit tensor. (translated by j___d)

The curve $y=y(x)$ satisfies $y'(0)=1.$ It satisfies the differential equation $(x^2 +9)y'' +(x^2 +4)y=0.$ Show that it crosses the $x$-axis between $$x= \frac{3}{2} \pi \;\;\; \text{and} \;\;\; x= \sqrt{\frac{63}{53}} \pi.$$

Assume that the system of differential equations $y'=-z^3$, $z'=y^3$ with the initial conditions $y(0)=1$, $z(0)=0$ has a unique solution $y=f(x)$, $z=g(x)$ defined for real $x$. Prove that there exists a positive constant $L$ such that for all real $x$, $$f(x+L)=f(x),\enspace g(x+L)=g(x).$$

Let $f(x)$ be a bounded continuous function on $[0,+\infty)$. Prove that every solutions of the equation $y''+14y'+13y=f(x)$ are bounded continuous functions on $[0,+\infty)$

Consider the function $y(x)$ satisfying the differential equation $y'' = -(1+\sqrt{x})y$ with $y(0)=1$ and $y'(0)=0.$ Prove that $y(x)$ vanishes exactly once on the interval $0< x< \pi \slash 2,$ and find a positive lower bound for the zero.

Let $f:[0,\infty)\to\mathbb R$ be differentiable and satisfy $$f'(x)=-3f(x)+6f(2x)$$for $x>0$. Assume that $|f(x)|\le e^{-\sqrt x}$ for $x\ge0$. For $n\in\mathbb N$, define $$\mu_n=\int^\infty_0x^nf(x)dx.$$ $a.$ Express $\mu_n$ in terms of $\mu_0$. $b.$ Prove that the sequence $\frac{3^n\mu_n}{n!}$ always converges, and the the limit is $0$ only if $\mu_0$.

Does there exist a twice differentiable function $f:\mathbb R\to\mathbb R$ such that $f'(x)=f(x+1)-f(x)$ for all $x$ and $f''(0)\ne0$? Justify your answer.

Find all solutions of the differential equation $zz" - 2z'z' = 0$ which pass through the point $x=1, z=1.$

Let $y_1$ and $y_2$ be two linearly independent solutions of the equation $$y''+P(x)y'+Q(x)=0.$$ Find the differential equation satisfied by the product $z=y_1 y_2$.

Given $x=\phi(u,v)$ and $y=\psi(u,v)$, where $ \phi$ and $\psi$ are solutions of the partial differential equation $$(1) \;\,\;\, \; \frac{ \partial \phi}{\partial u} \frac{\partial \psi}{ \partial v} - \frac{ \partial \phi}{\partial v} \frac{\partial \psi}{ \partial u}=1.$$ By assuming that $x$ and $y$ are the independent variables, show that $(1)$ may be transformed to $$(2) \;\,\;\, \; \frac{ \partial y}{ \partial v} =\frac{ \partial u}{\partial x}.$$ Integrate $(2)$ and show how this effects in general the solution of $(1)$. What other solutions does $(1)$ possess?

Show that if the differential equation $$M(x,y)\, dx +N(x,y) \, dy =0$$ is both homogeneous and exact, then the solution $y=y(x)$ satisfies that $xM(x,y)+yN(x,y)$ is constant.

Let $x: [0, \infty) \to\Bbb R$ be a differentiable function. Prove that if for all t>1 $$x'(t)=-x^3(t)+\frac{t-1}{t}x^3(t-1)$$ then $\lim_{t\to\infty} x(t) = 0$

Find all $c\in\mathbb R$ for which there exists an infinitely differentiable function $f:\mathbb R\to\mathbb R$ such that for all $n\in\mathbb N$ and $x\in\mathbb R$ we have $$f^{(n+1)}(x)>f^{(n)}(x)+c.$$

Let $u(t)$ be a continuous function in the system of differential equations $$\frac{dx}{dt} =-2y +u(t),\;\;\; \frac{dy}{dt}=-2x+ u(t).$$ Show that, regardless of the choice of $u(t)$, the solution of the system which satisfies $x=x_0 , y=y_0$ at $t=0$ will never pass through $(0, 0)$ unless $x_0 =y_0.$ When $x_0 =y_0 $, show that, for any positive value $t_0$ of $t$, it is possible to choose $u(t)$ so the solution is equal to $(0,0)$ when $t=t_0 .$

Let $g(t)$ and $h(t)$ be real, continuous functions for $t\geq 0.$ Show that any function $v(t)$ satisfying the differential inequality $$\frac{dv}{dt}+g(t)v \geq h(t),\;\; v(t)=c,$$ satisfies the further inequality $v(t)\geq u(t),$ where $$\frac{du}{dt}+g(t)u = h(t),\;\; u(t)=c.$$ From this, conclude that for sufficiently small $t>0,$ the solution of $$\frac{dv}{dt}+g(t)v = v^2 ,\;\; v(t)=c$$ may be written $$v=\max_{w(t)} \left( c e^{- \int_{0}^{t} |g(s)-2w(s)| \, ds} -\int_{0}^{t} e^{-\int_{0}^{t} |g(s')-2w(s')| \, ds'} w(s)^{2} ds \right),$$ where the maximum is over all continuous functions $w(t)$ defined over some $t$-interval $[0,t_0 ].$