Differential Equations of Calculus Problems- Calc I course

[tcking3]

So there are these 4 differential equations problems that are ruining my day. I can't find any videos or notes for differential equations of calculus which is for some reason in my Calc 1 class. They seem kind of simple but I just can't seem to get started... any help is appreciated!

1b)

When a condenser discharges electricity, the instantaneous rate of change of the voltage is proportional to the voltage in the condenser. Suppose you have a discharging condenser and the instantaneous rate of change of the voltage is1/100of the voltage (in volts per second). How many seconds does it take for the voltage to decrease by 90 % ?
1c)

A radium sample weighs1gram at time t = 0. At time t = 10 years it has diminished to 0.997 grams. How long will it take to diminish to 0.5grams?​
5d) Here is the solution of the differential equation:
y'[x] = (r y[x])/x2 with y[1] = a
Solution: a*er - r/x
Use derivative formulas and laws to explain this output.8) You put a tepid liquid refreshment into a very large refrigerator. The refrigerator is kept at a constant temperature S degrees. Let T[t] be the temperature of the liquid at time t minutes after the beverage is placed in the refrigerator. According to Newton's law of cooling, the rate of cooling T'[t] is proportional to the difference
T[t] - S ; that is, T'[t] = r (T[t] - S) = r T[t] - r S where r is a proportionality constant.

The following questions may be of some practical importance on a hot July afternoon. Suppose the refrigerator temperature S is held at 42 degrees (Fahrenheit), the original temperature of a bottle of a desirable beverage is 80 degrees , and the bottle of the beverage has cooled to 63 degrees after 10 minutes in the refrigerator. Plot the temperature of the bottle of the beverage as a function of time (in minutes). What will be the temperature of the bottle of the beverage after 30 minutes ? Approximately how many minutes will it take for the bottle of the beverage to cool to the refreshing temperature of 44 degrees ?

 

[I like Serena]

Hi tcking3! Welcome to MHB! (Smile)

Let's start with your first question.
Btw, generally we prefer to see a maximum of 2 questions per thread.

1b)
When a condenser discharges electricity, the instantaneous rate of change of the voltage is proportional to the voltage in the condenser. Suppose you have a discharging condenser and the instantaneous rate of change of the voltage is1/100of the voltage (in volts per second). How many seconds does it take for the voltage to decrease by 90 % ?

The voltage of a discharging condensor (same as a capacitor right?) is given by:
$$V(t) = V_0 e^{-t/\tau}$$
where $\tau$ is the so called characteristic time.

The instantaneous rate of change is its derivative.
So:
$$V'(0) = \frac {V_0}{100\text{ s}} $$

Can you find the characteristic time $\tau$ from these equations?

 

[tcking3]

Serena, thank you for your reply. I will limit my future inquiries to one or two questions. Could you get me started on #8 as I was able to figure out the first two with help from my TA this evening!

 

[I like Serena]

Serena, thank you for your reply. I will limit my future inquiries to one or two questions. Could you get me started on #8 as I was able to figure out the first two with help from my TA this evening!

Well, we ask that our users show their progress (work thus far or thoughts on how to begin). This way we can see where you are stuck or may be going astray and will be able to post the best help possible without potentially making a suggestion which you have already tried, which would waste your and our time.

Can you post what you have done so far?

 

[MarkFL]

Serena, thank you for your reply. I will limit my future inquiries to one or two questions. Could you get me started on #8 as I was able to figure out the first two with help from my TA this evening!

Can you completely state the IVP (Initial Value Problem, where you have an ODE (Ordinary Differential Equation) and one or more boundary conditions, which likely include an initial value)?

Once you have this, you want to obtain a general solution to the ODE and then plug in the initial values to determine the parameters in this solution, to obtain a particular solution that satisfies both the ODE and the boundary conditions?

And then once you have the particular solution you can use this to answer the questions. Note that you are asked to plot the temperature as it varies with time...this means you want to express the solution as temperature as a function of time. But notice also that you are asked how much time has elapsed before certain temperatures are reached...this means you will also want time as a function of temperature.

So, let's begin by formally stating the IVP...what do you get?

edit: I was not alerted that another reply had been given...ILS, please feel free to delete this post if you feel it's a "trampler." :D