Example 1  A 1500 gallon tank initially contains 600 gallons of water with 5 lbs of salt dissolved in it.  Water enters the tank at a rate of 9 gal/hr and the water entering the tank has a salt concentration of  lbs/gal.  If a well mixed solution leaves the tank at a rate of 6 gal/hr, how much salt is in the tank when it overflows?

Solution

First off, let’s address the “well mixed solution” bit.  This is the assumption that was mentioned earlier.  We are going to assume that the instant the water enters the tank it somehow instantly disperses evenly throughout the tank to give a uniform concentration of salt in the tank at every point.  Again, this will clearly not be the case in reality, but it will allow us to do the problem.

Now, to set up the IVP that we’ll need to solve to get Q(t) we’ll need the flow rate of the water entering (we’ve got that), the concentration of the salt in the water entering (we’ve got that), the flow rate of the water leaving (we’ve got that) and the concentration of the salt in the water exiting (we don’t have this yet).

So, we first need to determine the concentration of the salt in the water exiting the tank.  Since we are assuming a uniform concentration of salt in the tank the concentration at any point in the tank and hence in the water exiting is given by,

The amount at any time t is easy it’s just Q(t).  The volume is also pretty easy.  We start with 600 gallons and every hour 9 gallons enters and 6 gallons leave.  So, if we use t in hours, every hour 3 gallons enters the tank, or at any time t there is 600 + 3t gallons of water in the tank.

So, the IVP for this situation is,

This is a linear differential equation and it isn’t too difficult to solve (hopefully).  We will show most of the details, but leave the description of the solution process out.  If you need a refresher on solving linear first order differential equations go back and take a look at that section.

So, here’s the general solution.  Now, apply the initial condition to get the value of the constant, c.

So, the amount of salt in the tank at any time t is.

Now, the tank will overflow at t = 300 hrs.  The amount of salt in the tank at that time is.

Here’s a graph of the salt in the tank before it overflows.

Note that the whole graph should have small oscillations in it as you can see in the range from 200 to 250.  The scale of the oscillations however was small enough that the program used to generate the image had trouble showing all of them.

Example 2  A 1000 gallon holding tank that catches runoff from some chemical process initially has 800 gallons of water with 2 ounces of pollution dissolved in it.  Polluted water flows into the tank at a rate of 3 gal/hr and contains 5 ounces/gal of pollution in it. A well mixed solution leaves the tank at 3 gal/hr as well.  When the amount of pollution in the holding tank reaches 500 ounces the inflow of polluted water is cut off and fresh water will enter the tank at a decreased rate of 2 gallons while the outflow is increased to 4 gal/hr.  Determine the amount of pollution in the tank at any time t.

Solution

Okay, so clearly the pollution in the tank will increase as time passes.  If the amount of pollution ever reaches the maximum allowed there will be a change in the situation.  This will necessitate a change in the differential equation describing the process as well.  In other words, we’ll need two IVP’s for this problem.  One will describe the initial situation when polluted runoff is entering the tank and one for after the maximum allowed pollution is reached and fresh water is entering the tank.

Here are the two IVP’s for this problem.

The first one is fairly straight forward and will be valid until the maximum amount of pollution is reached.  We’ll call that time t_m.  Also, the volume in the tank remains constant during this time so we don’t need to do anything fancy with that this time in the second term as we did in the previous example.

We’ll need a little explanation for the second one.  First notice that we don’t “start over” at t = 0.  We start this one at t_m, the time at which the new process starts.  Next, fresh water is flowing into the tank and so the concentration of pollution in the incoming water is zero.  This will drop out the first term, and that’s okay so don’t worry about that. 

Now, notice that the volume at any time looks a little funny.  During this time frame we are losing two gallons of water every hour of the process so we need the “-2” in there to account for that.  However, we can’t just use t as we did in the previous example.  When this new process starts up there needs to be 800 gallons of water in the tank and if we just use t there we won’t have the required 800 gallons that we need in the equation.  So, to make sure that we have the proper volume we need to put in the difference in times.  In this way once we are one hour into the new process (i.e t - t_m = 1) we will have 798 gallons in the tank as required.

Finally, the second process can’t continue forever as eventually the tank will empty.  This is denoted in the time restrictions as t_e.  We can also note that t_e = t_m + 400 since the tank will empty 400 hours after this new process starts up.  Well, it will end provided something doesn’t come along and start changing the situation again.

Okay, now that we’ve got all the explanations taken care of here’s the simplified version of the IVP’s that we’ll be solving.

The first IVP is a fairly simple linear differential equation so we’ll leave the details of the solution to you to check.  Upon solving you get.

Now, we need to find t_m.  This isn’t too bad all we need to do is determine when the amount of pollution reaches 500.  So we need to solve.

So, the second process will pick up at 35.475 hours.  For completeness sake here is the IVP with this information inserted.

This differential equation is both linear and separable and again isn’t terribly difficult to solve so I’ll leave the details to you again to check that we should get.

So, a solution that encompasses the complete running time of the process is

Here is a graph of the amount of pollution in the tank at any time t.

Example 3  A population of insects in a region will grow at a rate that is proportional to their current population.  In the absence of any outside factors the population will triple in two weeks time.  On any given day there is a net migration into the area of 15 insects and 16 are eaten by the local bird population and 7 die of natural causes.  If there are initially 100 insects in the area will the population survive?  If not, when do they die out?

Solution

Let’s start out by looking at the birth rate.  We are told that the insects will be born at a rate that is proportional to the current population.  This means that the birth rate can be written as

where r is a positive constant that will need to be determined.  Now, let’s take everything into account and get the IVP for this problem.

Note that in the first line we used parenthesis to note which terms went into which part of the differential equation.  Also note that we don’t make use of the fact that the population will triple in two weeks time in the absence of outside factors here.  In the absence of outside factors means that the ONLY thing that we can consider is birth rate.  Nothing else can enter into the picture and clearly we have other influences in the differential equation.

So, just how does this tripling come into play?  Well, we should also note that without knowing r we will have a difficult time solving the IVP completely.  We will use the fact that the population triples in two week time to help us find r.  In the absence of outside factors the differential equation would become.

Note that since we used days as the time frame in the actual IVP I needed to convert the two weeks to 14 days.  We could have just as easily converted the original IVP to weeks as the time frame, in which case there would have been a net change of 56 per week instead of the 8 per week that we are currently using in the original differential equation.

Okay back to the differential equation that ignores all the outside factors.  This differential equation is separable and linear and is a simple differential equation to solve.  I’ll leave the detail to you to get the general solution.

Applying the initial condition gives c = 100.  Now apply the second condition.

We need to solve this for r.  First divide both sides by 100, then take the natural log of both sides.

We made use of the fact that  here to simplify the problem.  Now, that we have r we can go back and solve the original differential equation.  We’ll rewrite it a little for the solution process.

This is a fairly simple linear differential equation, but that coefficient of P always get people bent out of shape, so we’ll go through at least some of the details here.

Now, don’t get excited about the integrating factor here.  It’s just like  only this time the constant is a little more complicated than just a 2, but it is a constant!  Now, solve the differential equation.

Again, do not get excited about doing the right hand integral, it’s just like integrating !  Applying the initial condition gives the following.

Now, the exponential has a positive exponent and so will go to plus infinity as t increases.  Its coefficient, however, is negative and so the whole population will go negative eventually.  Clearly, population can’t be negative, but in order for the population to go negative it must pass through zero.  In other words, eventually all the insects must die.  So, they don’t survive and we can solve the following to determine when they die out.

So, the insects will survive for around 7.2 weeks.  Here is a graph of the population during the time in which they survive.

Example 4  A 50 kg mass is shot from a cannon straight up with an initial velocity of 10m/s off a bridge that is 100 meters above the ground.  If air resistance is given by 5v determine the velocity of the mass when it hits the ground.

Solution

First, notice that when we say straight up, we really mean straight up, but in such a way that it will miss the bridge on the way back down.  Here is a sketch of the situation.

Notice the conventions that we set up for this problem.  Since the vast majority of the motion will be in the downward direction we decided to assume that everything acting in the downward direction should be positive.  Note that we also defined the “zero position” as the bridge, which makes the ground have a “position” of 100.

Okay, if you think about it we actually have two situations here.  The initial phase in which the mass is rising in the air and the second phase when the mass is on its way down.  We will need to examine both situations and set up an IVP for each.  We will do this simultaneously.  Here are the forces that are acting on the object on the way up and on the way down.

Notice that the air resistance force needs a negative in both cases in order to get the correct “sign” or direction on the force.  When the mass is moving upwards the velocity (and hence v) is negative, yet the force must be acting in a downward direction.  Therefore, the “-” must be part of the force to make sure that, overall, the force is positive and hence acting in the downward direction.  Likewise, when the mass is moving downward the velocity (and so v) is positive.  Therefore, the air resistance must also have a “-” in order to make sure that it’s negative and hence acting in the upward direction.

So, the IVP for each of these situations are.

In the second IVP, the t₀ is the time when the object is at the highest point and is ready to start on the way down.  Note that at this time the velocity would be zero.  Also note that the initial condition of the first differential equation will have to be negative since the initial velocity is upward.

In this case, the differential equation for both of the situations is identical.  This won’t always happen, but in those cases where it does, we can ignore the second IVP and just let the first govern the whole process.

So, let’s actually plug in for the mass and gravity (we’ll be using g = 9.8 m/s² here).  We’ll go ahead and divide out the mass while we’re at it since we’ll need to do that eventually anyway.

This is a simple linear differential equation to solve so we’ll leave the details to you.  Upon solving we arrive at the following equation for the velocity of the object at any time t.

Okay, we want the velocity of the ball when it hits the ground.  Of course we need to know when it hits the ground before we can ask this.  In order to find this we will need to find the position function.  This is easy enough to do.

We can now use the fact that I took the convention that s(0) = 0 to find that c = -1080.  The position at any time is then.

To determine when the mass hits the ground we just need to solve.

We’ve got two solutions here, but since we are starting things at t = 0, the negative is clearly the incorrect value.  Therefore the mass hits the ground at t = 5.98147.  The velocity of the object upon hitting the ground is then.