0
votes
1answer
94 views

Unable to comprehend a connection between two equations

I was reading this paper and got stuck at the transition from Equation (13) to Equation (14) (p. 16/17). We got a function of the form: $y(t)=k(t)^{\alpha}h(t)^{\beta}$ We know it grows from zero ...
2
votes
0answers
16 views

Lipschitz continuity in two variables [duplicate]

Prove that $y \mapsto f(x,y)$ is Lipschitz continuous, where $$f(x,y) = \frac{y}{x} \ln{\frac{y}{x}}, \ \ \ |x-1| \leq \frac{1}{2}, |y-1| \leq \frac{1}{2}e$$ I tried to solve this, but I find it ...
0
votes
2answers
42 views

Absolute values in logarithms in a solution of differential equation

How have the moduli signs disappeared in the following step: $$\frac1{k}\left(\ln|g+kv| - \ln|g+ku|\right) = -t$$ Therefore $$ \ln\left(\frac{g+kv}{g+ku}\right) = -kt$$ $g$, $k$ and $u$ are ...
0
votes
1answer
45 views

function symmetric around a point

I need some quick help solving this: What is y(ln(2))if the function y satisfies $$\frac{dy}{dx}=1-y^2$$ and is symmetric about the point (ln(4),0)? I know that a function is symmetric about the ...
1
vote
1answer
52 views

$100(1.2)^t=\alpha t$. How do I solve for t?

I'm playing with a population model, and I wondered how to find the $t$ when the population goes to zero, with regard to $\alpha$. I've ended up with this, and I don't know how to solve for t. Can ...
1
vote
2answers
73 views

Solving equation with logarithms

I happen to use this heavy math for the first time for a long time (if ever) and don't know how to bite it. Given: $$\begin{align} A &= 1.45\\ B &= 4.1\\ C &= 14\\ ...
2
votes
3answers
57 views

Algebra help, logarithms, calculus

I'm seriously out of my depth. I have very basic understanding of logarithms and calculus. Please could someone walk me through how to get from: $$\frac{{d}p}{p} = -\frac{g}{R}\cdot ...
3
votes
1answer
128 views

Manipulating log function

I'm trying to find the value of a constant for $\ y(0) = 0$ in the following differential equation. $$\ 2\ln(2x+3y-1) - {2x+3y \over 2} = 2x+3y + k$$ Of course when plugging in the values, I get $\ ...
2
votes
1answer
385 views

Rearranging a logarithmic equation

I'm building a web app that displays the frequency of a sound. I have an equation that returns a pixel that a particular frequency should be mapped at. However I would like to reverse the equation so ...
1
vote
4answers
313 views

Finding all functions $f$ satisfying $f'(t)=f(t)+\int_a^bf(t)dt$

I am trying to find all functions f satisfying $f'(t)=f(t)+\int_a^bf(t)dt$. This is a problem from Spivak's Calculus and it is the chapter about Logarithms and Exponential functions. I gave up ...