All Questions
16
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Calculating Rate of Change and using differentials to project 3 years from now
Currently, BC is helping $R=5,000$ refugees. The number of refugees that BC must help is rising at a rate of $\frac{dR}{dt}=1,000$ refugees per year. Currently, the number of staff members is $N=100$ ...
- 785
1
vote
0
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41
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Sequence of Logic in Diffusion Problem DQ
Problem: If a tank is filled with 100 gallons of water and mistakenly added 300 pounds of salt. To fix the mistake the brine is drained at 3 gallons per minute and replaced with water at the same rate....
0
votes
1
answer
55
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Using an expression and an equation to get an ODE to describe something.
I have an expression and an equation, that I need to use to show that ODE describes something.
Let me put it into context
I have an expression for the Rate at Anti-Freeze flows $\mathcal{IN}$
and $\...
1
vote
1
answer
294
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Question in population dynamics using exponential growth rate equation
Given population doubles in 20 minutes, what is intrinsic growth rate r?
Attempt: Given population doubles, using exponential growth rate we have $\frac{dN}{dt}=2N$ so $N(t)=N_0e^{2t}$ therefore r=2, ...
- 395
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votes
1
answer
396
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Modelling interest with differential equations (IVP)
Problem : you set a bank account, with initial value k, the bank will pay you continuous interest of 12% per year.
a) write an initial value problem for your account balance y(t) after t years
Sol:
$$...
- 69
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votes
1
answer
263
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Mathematical expression for physical forces in pendulum ODE
A 16 lb weight is suspended from a spring having a spring constant of 5 lb/ft. Assume that an external force given by
24 sin (10t) and a damping force with damping constant 4, are acting on the spring....
- 612
1
vote
1
answer
425
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Finding the formula for T from Newton's Law of Cooling
I think I got a wrong answer because I skipped a particular step which seemed optional. I'm still not too sure what happened though and would appreciate your help...
Background:
Newton’s law of ...
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-1
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1
answer
103
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Calculating if Romeo and Juliet will stay together always or not. [closed]
I have two equations which describe the Love of Romeo for Juliet (R) and Love of Juliet for Romeo (J) as a function of time, $t$.
$R=-c_1e^{3t}-c_2e^{2t}$
$J=2c_1e^{3t}+c_2e^{2t}$
They will stay ...
- 25
4
votes
1
answer
1k
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Applications of First Order Differential Equations
Can I get help for this question please?
Suppose that a tank containing a liquid is vented to the air at the top and has an outlet at the bottom through which the liquid can drain. It follows from ...
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36
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Show that $y(t) = T(t − t_{0})$ also satisfies Newton’s law of cooling, $\frac{dT}{dt} = k(T_{e} − T)$, for any constant $t_{0}$.
This question if from MIT's Open Course Ware for Single Variable Calculus:
Show that $\:y(t) = T(t − t_{0})$ also satisfies Newton’s law of cooling, $\: \frac{dT}{dt} = k(T_{e} - T,)\:$for any ...
- 353
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votes
1
answer
684
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Finding the velocity, distance covered and terminal velocity of a skydiver
Here's the problem:
A skydiver and her equipment together weigh 192 pounds. Before the parachute is opened, there is an air drag force equal in magnitude to six times her velocity. Four seconds ...
6
votes
7
answers
10k
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How to prove $A\cos(\omega t-\phi)$ = $a\cos(\omega t)$ + $b\sin(\omega t)$ using $e^{i\theta}$?
I want to show that $A\cos\left(\omega t-\phi\right)$ = $a\cos\left(\omega t\right)$ + $b\sin\left(\omega t\right)$
First I verified for myself through the angle addition proof that:
$$ \cos\left(\...
- 363
2
votes
0
answers
116
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How might an applied mathematician view $ 1/x$, $\ln x$, and $e^x$?
I understand that the natural logarithm was developed by Gregoire de Saint-Vincent and Alphonse Antonio de Sarasa as to represent the area under the curve of the hyperbola $\frac1x$ before the ...
- 363
1
vote
0
answers
72
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Why does this mass equation not work in this calculus application problem?
This is a homework problem which I have already solved, but I want to know why my first approach did not work. I have already asked my professor this question, but he did not explain it very well.
...
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1
vote
4
answers
3k
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Modelling interest with differential equations (Interpretation)
I am having trouble interpreting the meaning of this differential equation model for interest on an account. The problem is as follows:
Assume you have a bank account that grows at an annual interest ...
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