midtlinjeg
2021-09-10
Answered

For which nonnegative integers n is

${n}^{2}\le n!?$

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asked 2022-09-08

Solve the differential equation $xyy\prime +xyy\prime ={y}^{2}+1$?

asked 2020-12-30

Solve differential equation$x{y}^{\prime}+3y=6{x}^{3}$

asked 2022-07-09

As part of my research I get the following differential equation. I need to solve for $\mathcal{V}(\gamma )$. In fact the requirement is not to solve but to show that $\mathcal{V}(\gamma )$ is monotonic in ${a}_{j}$$\mathrm{\forall}j$, (which I hope it is) where ${a}_{j}$ are positive valued constants which do not depend on $\gamma $. If it can be shown without solving the differential equation that is sufficient. Please provide some suggestions.

$\frac{{\textstyle \gamma}}{{\textstyle \mathrm{log}\left(e\right)}}\frac{{\textstyle d}}{{\textstyle d\gamma}}\mathcal{V}\left(\gamma \right)=1-\eta \left(\gamma \right)$

$\eta \left(\gamma \right)=\frac{{\textstyle 1}}{{\textstyle 1+\gamma \sum _{j}\frac{{\textstyle {a}_{j}}}{{\textstyle 1-{a}_{j}(-\gamma \eta \left(\gamma \right))}}}}$

where $j=\{1,\dots ,n\}$

$\frac{{\textstyle \gamma}}{{\textstyle \mathrm{log}\left(e\right)}}\frac{{\textstyle d}}{{\textstyle d\gamma}}\mathcal{V}\left(\gamma \right)=1-\eta \left(\gamma \right)$

$\eta \left(\gamma \right)=\frac{{\textstyle 1}}{{\textstyle 1+\gamma \sum _{j}\frac{{\textstyle {a}_{j}}}{{\textstyle 1-{a}_{j}(-\gamma \eta \left(\gamma \right))}}}}$

where $j=\{1,\dots ,n\}$

asked 2022-09-07

What is a solution to the differential equation $\frac{dy}{dt}=\frac{{e}^{t}}{4y}$?

asked 2022-04-10

For every differentiable function $f:\mathbb{R}\to \mathbb{R}$, there is a function $g:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ such that $g(f(x),{f}^{\prime}(x))=0$ for every $x$ and for every differentiable function $h:\mathbb{R}\to \mathbb{R}$ holds that

being true that for every $x\in \mathbb{R}$, $g(h(x),{h}^{\prime}(x))=0$ and $h(0)=f(0)$ implies that $h(x)=f(x)$ for every $x\in \mathbb{R}$.

i.e every differentiable function $f$ is a solution to some first order differential equation that has translation symmetry.

being true that for every $x\in \mathbb{R}$, $g(h(x),{h}^{\prime}(x))=0$ and $h(0)=f(0)$ implies that $h(x)=f(x)$ for every $x\in \mathbb{R}$.

i.e every differentiable function $f$ is a solution to some first order differential equation that has translation symmetry.

asked 2022-06-21

I have a differential equation of the form

$dy/dx+p(x)y=q(x)$

under the condition that $q(x)=300$ if $y<3312$ and $q(x)=0$ if $y\ge 3312$.

I could not understand how to solve this differential equation with such heavy side function ?

Any hints?

$dy/dx+p(x)y=q(x)$

under the condition that $q(x)=300$ if $y<3312$ and $q(x)=0$ if $y\ge 3312$.

I could not understand how to solve this differential equation with such heavy side function ?

Any hints?

asked 2021-03-08

Solve differential equation ${y}^{\prime}+y=x,\text{}y\left(0\right)=1$