All Questions
Tagged with ordinary-differential-equations exponential-function
235
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Modelling Population and Differential Equations
A population of a species of animal had the following differential equation:
$${dP\over dt} = rP \left(1-{P\over K}\right)$$
$P$ was the population.
$t$ was the time in years.
$K$ was the carrying ...
- 95
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2
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720
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Linear first order ODE with exponential coefficient
I have a problem. I have to solve a linear first order, homogeneous ODE with an exponential coefficient.The equation is as follows;
$\frac{dP(x)}{dt}-e^{-ax}P(x)=0$.
Clearly, its impossible to find ...
- 35
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1
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$e^{pt}$ or $(1+p)^t$ What is the difference in modeling exponential growth and decay?
I would like to better understand, when the function $e^{pt}$ is the "better" choice and when (if at all) $(1+p)^t$ should be used.
To give a classic example: Say we want to model radioactive decay ...
- 75
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Alternate proof for the derivative of $e^x$ using L'Hospital's rule – is there a generalization?
Before you say you can't use the derivative of a function to prove the derivative of a function, just please see the proof.
I created it myself, and was wondering whether this could applied to any ...
- 446
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2
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48
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Find the range of $y$ in a DE
Consider the equation $$y' = y^2 - y - 2 = (y+1)(y-2).$$ If $y(10) = 0$, find the range of $y(t)$ for $t>10$. That is, find the best $A$ and $B$ such that $A<y(t)<B$ for $t>10$.
From ...
- 329
2
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1
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730
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Interval of Validity of a Solution
By separation of variables, the general solution to $y' = 3y - y^2$ is
$y(t) = 0$ or $y(t) = \frac{3}{1+ce^{-3t}}$.
Find the interval of validity of the solution satisfying $y(0) = 6$.
(i.e. find $a$ ...
- 329
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Can we express all functions in the exponential family with this differential equation?
Using the "prime" notation for differentiation $$f'(x) = \frac{\partial }{\partial x}\{f\}(x)\\f''(x) = \frac{\partial^2 }{\partial x^2}\{f\}(x)\\\vdots\\f^{(k)}(x) = \frac{\partial^k }{\partial x^k}\{...
- 26.1k
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59
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For which input signals is the response resonant?
Consider the differential equation $y''' + 5y'' + 8y' + 6y = f(x)$.
For which following input signals $f(x)$ is the response resonant? Choose all those that apply.
$e^x, e^{-x}, e^{3x}, e^{-3x}, e^x\...
- 4,581
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345
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Exponential Decay with a Factory Machine
The value V of a factory machine depreciated with time t years such that dV/dt=-kV, for some constant k>0.
(i) Show that V=V_0 e^(-kt) satisfies the given differential equation.
(ii) The initial ...
- 29
1
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2
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63
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Cauchy problem: $x'=\frac{x}{t^2+1}, x(0)=1$.
$$x'=\frac{x}{1+t^2}, \qquad x(0)=1$$
I know the solution to the problem, but I don't get to the right solution myself. My solution:
$$\int \frac{dx}{x}=\int \frac{dt}{1+t^2}$$
$$\ln(x)=\cot(t)+ C$...
- 1,412
1
vote
1
answer
222
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Solution for differential equation describing linear input and exponential decay
I try to model the concentration over time when there's an linear input and an exponential output (i.e. exponential decay) with known half-life.
Input(t) = a+b*t
...
- 13
1
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1
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114
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Differential equation $x'(t) e^{-x'(t)^2} = c$ with Lambert W function.
Let $x(t)$ be a smooth function, find a solution of $x'(t) e^{-x'(t)^2} =c$.
I first saw this DE in a question of
Frederic Chopin (Integral involving piecewise continuous function), when I started ...
- 824
1
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3
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62
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Why is $e^{\int_0^t A(s)} \mathrm{d} s$ a solution of $x' = Ax$ iff all the entries of $A(s)$ are constant?
I have seen this result in a few places on the internet and I am trying to prove it, but have found no way to start. I am trying to use the matrix $B = e^{\int_0^t A(s)} \mathrm{d} s$ and expand it ...
- 1,965
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How to prove this inequality with Big O term?
Let $s= s_0-\zeta_0^{-1/2}b^{-1/6}(1+\mathcal{O}(\sqrt{b}))$
where $\zeta_0 = \left(\frac{3\pi}{4}\right)^{2/3}$ and $s_0 = b^{-2/3}\zeta_0.$ Note that here $b$ is a parameter.
We define $\omega(s) = ...
- 9,258
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5
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What is the derivative of a function of the form $u(x)^{v(x)}$?
So I have a given lets say $(x+1)^{2x}$ in addition to $\frac{\mathrm dy}{\mathrm dx}a^u=a^u\log(a)u'$. I still have to multiply this by the derivative of the inside function $x+1$ correct?
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