All Questions
Tagged with ordinary-differential-equations exponential-function
235
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Tan of sum using differential equations
Define $S(x)$ and $C(x)$ as exponential functions, and $T(x) = S(x)/C(x)$. We want to derive the formula for $T(x+y)$, but the exercise I'm working on suggests setting up a differential equation $g(T(...
- 133
3
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1
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309
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Recommendations about the exponential function
I am studying differential equations and I am very surprised by how omnipresent the exponential function is. It pops up everywhere, but there isn't usually a lot of detail provided in introductory ...
- 167
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229
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Solving $\frac{dN}{dt} = rN$ (population growth equation) question about using integration to solve simple separable differential equation
I covered differential equations a long time ago and so was confused when my Population Ecology textbook displayed the following "proof" to solve $\frac{dN}{dt} = rN$ for $N$.
My (probably ...
- 887
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1
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The exponential term in a differential equation
Given the differential equation
$P = \frac{L}{De^{-mt} + 1}$,
an intial population of 1000, $L = 10000$ and $m = 0.003$, find the number of years it will take for the population to triple.
To solve ...
- 41
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Family of straight lines tangent to $e^x$
I just started to learn about Differential equations, I've been reading about families of curves, but I I just got stuck with this problem. I have been Trying to solve it, I have come to find the line ...
- 11
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Don't understand how a power of e ends up as a factor in from of e
I have a very basic issue that I can't seem to wrap my head around: I am trying to solve the membrane equation
$$\tau \frac{dV(t)}{dt} = -V(t) + E(t)$$
With a time constant $\tau$, the voltage $V(t)$ ...
- 1
5
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2
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Exponential growth of a cow farm with constraints in Minecraft
This question is distinct from Exponential growth of cow populations in Minecraft in that an important constraint present in Minecraft is missing from that post. Here are the following constraints:
...
- 861
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How can I handle$~\exp(\ln|x|)~$to solve 1st order linear DE?
RHS and LHS are same.
$$\exp\left(\ln\left(x\right)\right)=\exp\left(\ln\left(x\right)\right)\tag{1}$$
Taking log.
$$\ln\left(\exp\left(\ln\left(x\right)\right)\right)=\ln\left(\exp\left(\ln\left(x\...
- 1,519
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if $\dfrac{d\phi}{dx} \leq k \phi$, $\phi(0) = 0$ then $\phi(x) \equiv 0$.
I need a hint (not a full solution, please) on how to prove the following result:
Prove that if $\left\{ \begin{array}{l} \dfrac{d \phi}{dx} \leq k \cdot \phi\\
\phi(0)=0\end{array}\right., k>0$, ...
- 303
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71
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Alternative argument to show that function diverges everywhere
Consider the function:
$$f(x) = x^{\frac{1}{2}} + \frac{1}{2}x^{-\frac{1}{2}} +\frac{2}{3}x^{\frac{3}{2}} - \frac{1}{4}x^{-\frac{3}{2}} +\frac{1}{15}x^{\frac{5}{2}} + \cdots$$
which is constructed ...
- 2,466
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1
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71
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Intuitively, why does growth proportional to the population size not diverge but growth proportional to pairs diverges to infinity in finite time?
It's interesting that growth which is proportional to the current population doesn't diverge to infinity, while growth that is proportional to higher powers of the current population.
That is, the ...
- 1,489
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Simplification of series (A good approximation will also work)
I have solved a system of linear ODEs and obtained a solution of the form
$
y_{ij}(t)=\sqrt{\frac{\gamma^{i(i-1)}}{\gamma^{j(j-1)}}}(\frac{-2\epsilon}{\alpha})^{i-j} \sum_{h=j}^i\frac{e^{-\frac{\alpha ...
- 61
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1
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202
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Radio active decay formula is not growth formula, $p = p_0 * e^{kt}$, what is the formula for radio active decay, and why?
Trying to solve this question:
A radioactive material is known to decay at a yearly rate proportional to the amount at each moment. There were 2000 grams of the material 10 years ago. There are 1990 ...
- 671
2
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1
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58
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Exponential populations that depend upon each other
I have a question about how to solve an exponential problem that involves two populations, each which depends on the other.
For example, let's say we have an initial population of $h$ humans that ...
- 153
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1
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62
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Why does this data not line up with the differential equation that's supposed to model it?
Sorry for the bad title, I wasn't sure how to ask this specific question.
So for a (extra credit) homework assignment, I wrote a python program for my differential equations class that should model ...
