Differential Equations Problems And Solutions Pdf

[Jordan Tucker]

Weare now going to start looking at nonlinear first order differential equations. The first type of nonlinear first order differential equations that we will look at is separable differential equations.

Note that in order for a differential equation to be separable all the \(y\)'s in the differential equation must be multiplied by the derivative and all the \(x\)'s in the differential equation must be on the other side of the equal sign.

We will also have to worry about the interval of validity for many of these solutions. Recall that the interval of validity was the range of the independent variable, \(x\) in this case, on which the solution is valid. In other words, we need to avoid division by zero, complex numbers, logarithms of negative numbers or zero, etc. Most of the solutions that we will get from separable differential equations will not be valid for all values of \(x\).

So, we now have an implicit solution. This solution is easy enough to get an explicit solution, however before getting that it is usually easier to find the value of the constant at this point. So apply the initial condition and find the value of \(c\).

Recall that there are two conditions that define an interval of validity. First, it must be a continuous interval with no breaks or holes in it. Second it must contain the value of the independent variable in the initial condition, x = 1 in this case.

To finish the example out we need to determine the interval of validity for the solution. If we were to put a large negative value of \(x\) in the solution we would end up with complex values in our solution and we want to avoid complex numbers in our solutions here. So, we will need to determine which values of \(x\) will give real solutions. To do this we will need to solve the following inequality.

The quadratic will be zero at the two points \(x = 2 \pm 2\sqrt 2 \). A graph of the quadratic (shown below) shows that there are in fact two intervals in which we will get positive values of the polynomial and hence can be possible intervals of validity.

From the graph of the quadratic we can see that the second one contains \(x\) = 5, the value of the independent variable from the initial condition. Therefore, the interval of validity for this solution is.

It is not possible to find an explicit solution for this problem and so we will have to leave the solution in its implicit form. Finding intervals of validity from implicit solutions can often be very difficult so we will also not bother with that for this problem.

Since the algorithms used to solve are adaptive, the size of solutions might be different. Now, if I say that all my growth rates of species have the units hour^(-1) and all other constants have appropriate units and specify the time span to be 0.0 to 100.0 and I wish to track the population size in each hour. Given the model equations, if the length of solution comes out be 1000, then:

So from I understand, if I set up the time span to be 0.0 to 10.0 and my units are still hour^(-1) and the algorithm somhow calculates stepsize as 0.5 (assume, I dont know if its possible or not), I must get the solution vector to be of length 10.0/0.5 i.e., 20, right?

The time points the solver will use will probably not be evenly spaced. They will depend on how fast things are changing in your simulation. I would encourage you to try using the interpolation - it is a good feature.

I am quiet new to solving ODEs, rather this is my first time.

Can you shed some light upon the interpolating function as I did not find any good material on it online?

Is it anything like the integrated function of the ODE/s?

Why do I need to learn to do complex math operations on paper when most can be done automatically in software like Maple. For instance, as long as I learn the concept and application for how aspects of linear algebra and differential equations work, won't I be able to enter the appropriate info into such a software program and not have to manually do the calculations?

Just to clarify, I'm not trying to offend any mathematicians or to belittle the importance of math. From CS I recognize that knowing the deep details of an algorithm can be useful, but that is equally important to be able to work abstractly. Just trying to get some perspective on how to approach the next few years of study.

Both. One is difficult without the other. How are you going to solve equations that Maple can't solve? How are you going to solve it, exactly or numerically? What's the best way to solve something numerically? How can you simplify the problem to get an approximate answer? How are you going to interpret Maple's output, and any issues you have with its solution? How can you simplify the answer it gives you? What if you are only interested in the problem for a particular set of values/parameters/in a particular range? What happens if a parameter is small? How many solutions are there? Does a solution even exist?

Using a CAS without knowing the background maths behind the problems you're trying to solve is like punching the buttons on a calculator without knowing what numbers are, what the operations mean or what the order of operations might be.

I will second Bennett, the point is both. Consider the analogy that learning mathematics and physics is much like constructing maps. First, you will see maps others have created, how details are crafted, norms, what are usual rules, what are the great maps for certain regions. This is the highview.

However, you must be sure these maps are correct. Therefore, you'll go to the places they give you directions to and check if it matches. This is the ground level. You have to make sure you are following instructions correctly, arriving at the same results, be able to walk yourself through the path.

It's the only way you have a firm, solid, sharp knowledge of anything you study. Learning how to switch between the bird's-eye view and sniffing the ground is part of the apprenticeship of anyone in science.

Without knowing the details of a process, it is extremely difficult to program tools yourself that compute this process. Put more succinctly, without understanding an algorithm, it is nearly impossible to implement the algorithm. This is not nearly its only justification, but I would wager it is the most relevant, given your background.

Unlike mathematicians of long ago, I don't have to look up logarithm tables whenever I need to calculate $\log(24)$. My calculator or computer can do that for me, and that's a great advantage. They are great tools. Likewise, my tools can tell me what $298379187912 / 81238.235$ is, or whether or not $e^9$ is greater than $3^8$.

What these operations have in common is that they are simple, mechanical operations. Solving a differential equation is far more complicated, especially when it comes to PDEs. That doesn't mean computers can't help you solve some of them, but observe that a numerical solution is quite different from an analytical solution, and the latter can provide added information that the numerical solution cannot. There are also still a lot of PDEs that we simply do not know how to solve analytically (yet).

For instance. Why learn to do basic math, when calculators are so omnipresent? Well, there was a situation a few years back where the register had gone down, so the cashier was having to add up the amounts 'by hand' using a hand calculator. This one lady further up the line got up to the register and when the cashier added it up, she objected to the amount.

Why? Well, because she could do basic math in her head. The cashier couldn't. So the cashier just kept typing in the quantities, and the same amount came out, so the cashier was just trusting the calculator. Long story short - the battery on the calculator was going and who knows how long it had been spitting out bad data and customers had been getting charged the wrong amount. The manager apologized, batteries were replaced.

Same thing for your Differential Equations. You need to know how DifEQs work so that when you do some little problem somewhere in setting up Maple to solve it for you, and don't realize you have a setting off, and it spits out a very wrong answer ... you know enough to sanity-check the result that Maple gives you.

You're going to need to know how to solve differential equations, especially if you're interested in quantum mechanics. Even some of the most basic examples of potentials used for quantum mechanics make CAS choke.

Examples: Delta potential, finite potential well (a lot easier to use the basic theory of differential equations to get a transcendental equation and use a root finding algorithm), and the quantum harmonic oscillator (look up the ladder operator method, really cool way of looking at problems like this).

Quantum Mechanics problems cannot yet be solved automatically on the computer. We're not even close to that point. There are many computational theories and approximations, but no canned program that can reliably give you solutions to a wide range of Quantum problems. Not even narrow ranges of problems can be solved so easily at this point. Maybe you'll be someone who makes significant contributions to computing, but you won't get there in Quantum Mechanics without knowing a tremendous amount of theory; this is because intuition in Quantum doesn't come from personal experience so much as it does through Math.

The computer architecture was designed by one of the most brilliant Mathematicians of the 20th century specifically to tackle non-linear problems, and the basic architecture hasn't significantly changed since. Great strides have been made in solving non-linear problems, but much of Quantum remains an enigma. By the way, it was this same Mathematician who invented the computer architecture who proposed the rigorous Mathematical framework for Quantum Mechanics that we still use today.

If you're interested in research, you'll have one type of career. If you interested in using existing tools to solve problems, you'll have another type of career. Each is equally valid, and it comes down to personal preference, talent, temperament, resources, etc.. Significant advances can be found in any field, but not usually by those who end up doing something they really don't like.

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