Need some advice -- Studying oscillations before differential equations?

[EnricoHendro]

Hello there,
I need some advice here. I am currently studying intro physics together with calculus. I am currently on intro to oscillatory motion and waves (physics-wise) and parametric curves (calc/math-wise). I noticed that in the oscillatory motion section, I need differential equation. Should I skip right to differential equation so that I can resume my physics study along with calc 2 and 3, or should I put my physics on hold and study the calc 2 first and then calc 3 and then linear algebra and then differential equation??

 

[jedishrfu]

No continue what you are doing. It's never good to break your stride and get side-tracked. The usual college scheme is to run introductory physics in parallel with or one semester off from your math sequence.

semester 1: Calc 1
semester 2: Calc 2 + Physics 1
semester 3: Calc 3 + Physics 2
semester 4: Diff Eqns + Modern Physics (QM + Spec Relativity)
semester 5: Linear Algebra + Formal EM Theory
semester 6: Statistics or Stat Thermo which will cover it too.

so basically you math is one semester ahead of when you'll really need it. This was how I was roughly taught.

It's best to make a todo list with some reference of what you want to go back to later and plow ahead. Basically, it's good to know what you don't know and then fix it when you can.

 

[symbolipoint]

Your introductory Physics course might make some use of differential equations but they are not very complicated at that level.

 

[MidgetDwarf]

At your level, it would mostly solving simple ODE by the method of separation of variables.

No need to read an actual book on ODE. The most important thing is that you what derivatives are (ie., what are you taking the derivative with respect to..), and what integration is (what are you integrating with respect too).

 

[EnricoHendro]

No continue what you are doing. It's never good to break your stride and get side-tracked. The usual college scheme is to run introductory physics in parallel with or one semester off from your math sequence.

semester 1: Calc 1
semester 2: Calc 2 + Physics 1
semester 3: Calc 3 + Physics 2
semester 4: Diff Eqns + Modern Physics (QM + Spec Relativity)
semester 5: Linear Algebra + Formal EM Theory
semester 6: Statistics or Stat Thermo which will cover it too.

so basically you math is one semester ahead of when you'll really need it. This was how I was roughly taught.

It's best to make a todo list with some reference of what you want to go back to later and plow ahead. Basically, it's good to know what you don't know and then fix it when you can.

I see. It’s a relieve to know I am still on the right track. Thank you for your advice. What did physics 1 cover in your uni?? I am a self taught so I don’t know what physics 1 covers what physics 2 covers and so on. I am using physics for scientists and engineers by serway, and a youtuber said that she used that book on her intro physics class

 

[EnricoHendro]

Your introductory Physics course might make some use of differential equations but they are not very complicated at that level.

I see, so I should be good to continue my intro physics? Am I getting this right?

 

[EnricoHendro]

At your level, it would mostly solving simple ODE by the method of separation of variables.

No need to read an actual book on ODE. The most important thing is that you what derivatives are (ie., what are you taking the derivative with respect to..), and what integration is (what are you integrating with respect too).

I see. I am familiar with separation of variables and derivatives and integrals. But, What is ODE??

 

[MidgetDwarf]

ODE stand for Ordinary Differential Equations. You run into them in the physics derivations in the intro physics courses. In more advanced courses in Physics, you apply methods of ODE to solve actual problems.

I had misread your post, thinking you wanted to study ODE. My error.

If you want a clear and straight to the point Calculus text, which gives you a why, and is not bogged down by colorful seizure inducing images on every page. Take a look at Thomas: Calculus with Analytical Geomtry 3rd ed. It has to be this edition. Later books are not so great.

Since you are self studying. I highly recommend Alonso and Finn: Fundamental University Physics. It actually makes use of the Calculus. It is straight and too the point. Have a look at it online to see if you like it.

Anyhow. Make sure to do as many problems as you can in your calculus and physics studies. S

 

[EnricoHendro]

ODE stand for Ordinary Differential Equations. You run into them in the physics derivations in the intro physics courses. In more advanced courses in Physics, you apply methods of ODE to solve actual problems.

I had misread your post, thinking you wanted to study ODE. My error.

If you want a clear and straight to the point Calculus text, which gives you a why, and is not bogged down by colorful seizure inducing images on every page. Take a look at Thomas: Calculus with Analytical Geomtry 3rd ed. It has to be this edition. Later books are not so great.

Since you are self studying. I highly recommend Alonso and Finn: Fundamental University Physics. It actually makes use of the Calculus. It is straight and too the point. Have a look at it online to see if you like it.

Anyhow. Make sure to do as many problems as you can in your calculus and physics studies. S

Ah I see. I thought you meant open dynamics engine. Ah, thomas calculus, I have one at home. But it is translated to my native language (indonesia) and I found the translated version hard to understand. I am taking paul’s online notes (recommended to me by a member in this forum a long time ago). I see, I will check the finn and alonso one. Yes, I always make sure I do all the problems I can find in my textbook. Thank you for your advice

 

[jedishrfu]

Physics 1 covered the basics, frames of reference, statics and dynamics. Physics 2 covered electricity and magnetism. Many departments used Haliday and Resnick of RPI as the intro book.

It’s still being updated and used today with many more sidebars and pictures to make it interesting and perhaps to distract the student from the fear of physics and math.

Modern Physics covered QM and Special Relativity at an intro level. We studied the Bohr atom, the square well and the Relativity paradoxes too.

Checkout Openstax.org as they have some good quality Physics and Calculus books.

 

[jtbell]

At your level, it would mostly solving simple ODE by the method of separation of variables.

Not even that fancy. At the level of Serway or Halliday/Resnick, the goal is simply to introduce the concept of a differential equation. IIRC they simply give you the basic solution and plug it into the d.e. to show that it works, and maybe evaluate the constants.

For many years I taught an "intro to modern physics" course which included solving Schrödinger's equation for the "particle in a box". Most students knew some calculus but had not taken a differential equations course yet. For that, we did introduce the method of separation of variables. Some students even needed to be introduced to the idea of partial derivatives first.

Then, after we had the separated d.e.'s in x and t, our method was to guess a solution (with some hints from me) based on functions that they knew about (sin, cos, exponential, etc.), plug it in and see if it works. If it doesn't work, observe how it fails and use that information to guess another candidate solution.

I told them, "this obviously isn't rigorous, but you'll eventually learn the rigorous methods in your differential equations course."

 

[mpresic3]

If you are taking two college courses simultaneously and are tested in the material, you probably do not have the luxury of delaying your learning of oscillations, until after being introduced to differential equations. If you are self studying, there is nothing wrong with learning differential equations first. If you are in math and physics courses as an undergraduate, this may be the first time, but it won't be the last time you will be called upon to confront the problem that you may not be able to be introduced to all the math prerequisites for the physics. Many other physicists "learned math as they went forward". This is a hazard of the profession. You may need to check with your teaching assistant or professor for the best way forward. One trick, I learned when I studied oscillations was the following. I did not like the fact that in oscillations, the textbooks required us to guess the solution to the differential equation. Suppose the reader is not very good at guessing. Instead, I realized that I already was introduced to conservation of energy in my physics courses. I also realized velocity was dx / dt, I could write the total energy E as the sum of kinetic energy: KE = 0.5 m ( dx/dt ) squared, and the potential energy PE = 0.5 k (x) squared. Then I solved for the velocity v in terms of the total energy (constant) E and the variable x. Then I came up with a first order differential equation dx / dt = square root ( ( 2 / m ) times ( E - 0.5 k x squared ) ). then put the equation in the form: dt = dx / square root ( ( 2 / m ) times ( E - 0.5 k x squared ) ), or t = integral (dx / square root ( ( 2 / m ) times ( E - 0.5 k x squared ) ). This integral can be solved analytically, sooner you see this in your basic calculus class before being introduced to differential equations (the solution is a inverse sine). Then you take the sines of both sides and eventually you get x = and equation of the form A cos w t. with constants A and w that are functions of the constants k, and m and energy E. Hope this helps. This is a lot more steps than guessing a solution but it does not rely on guesswork, and it uses calculus before differential equations.

Source https://www.physicsforums.com/threa...-differential-equations.1002250/#post-6483466

 

[EnricoHendro]

Physics 1 covered the basics, frames of reference, statics and dynamics. Physics 2 covered electricity and magnetism. Many departments used Haliday and Resnick of RPI as the intro book.

It’s still being updated and used today with many more sidebars and pictures to make it interesting and perhaps to distract the student from the fear of physics and math.

Modern Physics covered QM and Special Relativity at an intro level. We studied the Bohr atom, the square well and the Relativity paradoxes too.

Checkout Openstax.org as they have some good quality Physics and Calculus books.

I see. This will help me a lot in designing my study plan for physics and calculus. I am going to follow your "syllabus". Thank you for the link. I used Openstax when I first started studying physics. They are easy to understand. Will come back to that.

 

[EnricoHendro]

Not even that fancy. At the level of Serway or Halliday/Resnick, the goal is simply to introduce the concept of a differential equation. IIRC they simply give you the basic solution and plug it into the d.e. to show that it works, and maybe evaluate the constants.

For many years I taught an "intro to modern physics" course which included solving Schrödinger's equation for the "particle in a box". Most students knew some calculus but had not taken a differential equations course yet. For that, we did introduce the method of separation of variables. Some students even needed to be introduced to the idea of partial derivatives first.

Then, after we had the separated d.e.'s in x and t, our method was to guess a solution (with some hints from me) based on functions that they knew about (sin, cos, exponential, etc.), plug it in and see if it works. If it doesn't work, observe how it fails and use that information to guess another candidate solution.

I told them, "this obviously isn't rigorous, but you'll eventually learn the rigorous methods in your differential equations course."

yeah, Serway does give us the basic solution and then proof it backwards (they show you the general solution first and then derivate it twice to get to the second order diff equation). This actually what led me to post this thread.

 

[EnricoHendro]

If you are taking two college courses simultaneously and are tested in the material, you probably do not have the luxury of delaying your learning of oscillations, until after being introduced to differential equations. If you are self studying, there is nothing wrong with learning differential equations first. If you are in math and physics courses as an undergraduate, this may be the first time, but it won't be the last time you will be called upon to confront the problem that you may not be able to be introduced to all the math prerequisites for the physics. Many other physicists "learned math as they went forward". This is a hazard of the profession. You may need to check with your teaching assistant or professor for the best way forward. One trick, I learned when I studied oscillations was the following. I did not like the fact that in oscillations, the textbooks required us to guess the solution to the differential equation. Suppose the reader is not very good at guessing. Instead, I realized that I already was introduced to conservation of energy in my physics courses. I also realized velocity was dx / dt, I could write the total energy E as the sum of kinetic energy: KE = 0.5 m ( dx/dt ) squared, and the potential energy PE = 0.5 k (x) squared. Then I solved for the velocity v in terms of the total energy (constant) E and the variable x. Then I came up with a first order differential equation dx / dt = square root ( ( 2 / m ) times ( E - 0.5 k x squared ) ). then put the equation in the form: dt = dx / square root ( ( 2 / m ) times ( E - 0.5 k x squared ) ), or t = integral (dx / square root ( ( 2 / m ) times ( E - 0.5 k x squared ) ). This integral can be solved analytically, sooner you see this in your basic calculus class before being introduced to differential equations (the solution is a inverse sine). Then you take the sines of both sides and eventually you get x = and equation of the form A cos w t. with constants A and w that are functions of the constants k, and m and energy E. Hope this helps. This is a lot more steps than guessing a solution but it does not rely on guesswork, and it uses calculus before differential equations.

Source https://www.physicsforums.com/threa...-differential-equations.1002250/#post-6483466

oh I see. Yeah, I don't actually want to delay my study on oscillations and physics as a whole. I always thought physicists kind of have all the necessary math (like they don't have to learn the math as they move forward). Thank you for sharing your trick I will try to digest your method.

 

[vela]

I did not like the fact that in oscillations, the textbooks required us to guess the solution to the differential equation. Suppose the reader is not very good at guessing.

I remember not being satisfied with guessing either when I took intro physics. In my classes, I will acknowledge that some students might feel the same way and spend a little time showing how to solve the differential equation by trying $e^{rt}$. I think it's worth the effort for intro physics students to learn how to solve $\ddot x(t) + \omega^2 x = 0$. They'll see the same DE again later in quantum. Plus, it gives me the opportunity to remind them of or introduce them to Euler's formula and complex numbers, which is helpful later on when they cover interference and diffraction and certainly when they learn about quantum mechanics.

 

[hutchphd]

yeah, Serway does give us the basic solution and then proof it backwards (they show you the general solution first and then derivate it twice to get to the second order diff equation). This actually what led me to post this thread.

When it comes to solving Differential Equations I suggest an attitude of "all is fair" including guessing (or believing the book solution) and testing it out by substitution. If you cannot perhaps generate the solution yourself, just understand it as a gift for now. Becoming adept in that art comes only with repeated attempts and learning the Physics will aid you in obtaining that skill.

 

[mpresic3]

I notice that the procedure using conservation of energy uses a lot more steps than guessing a solution. The moral I have heard over and over, is that learning advanced mathematical tools makes physics, easier not harder, to learn. It is easier for most students to learn (or teachers to teach) calculus based physics than algebra based physics.

I grant that some comments that followed my earlier posts highlighted the importance of the solve by intuitive guess method. I agree, the second order DE will be met repeatedly in other contexts. In addition, physicist should develop their intuition for guessing solutions, a la Einstein or Feynman.

However, the conservation of energy method also introduces techniques that will be seen in the future, similar to Hamilton - jacobi method. If time permitted, and I was intructor for the class, I would present both

 

[hutchphd]

By all means: the more more you know, the more you know. And the total greatly outweighs the sum of the parts. The only real question is the optimal way to proceed. I would have preferred to know calculus before learning the physics but in some ways physics was much easier for me and it helped me learn calculus.
Being a student bestows the wonderful liberty to unabashedly ask questions.

 

[onatirec]

spend a little time showing how to solve the differential equation by trying $e^{rt}$. I think it's worth the effort for intro physics students to learn how to solve $\ddot x(t) + \omega^2 x = 0$.

This is a very useful technique in vibrations, and can be used again to derive solutions for characteristic equations with damping, etc.

Then I solved for the velocity v in terms of the total energy (constant) E and the variable x. Then I came up with a first order differential equation dx / dt = square root ( ( 2 / m ) times ( E - 0.5 k x squared ) ). then put the equation in the form: dt = dx / square root ( ( 2 / m ) times ( E - 0.5 k x squared ) ), or t = integral (dx / square root ( ( 2 / m ) times ( E - 0.5 k x squared ) ). This integral can be solved analytically

Honestly, I find this kind of convoluted - and trying to deal with an integral that needs to be solved analytically is very unlikely to help you if you get stuck on a test.

I'd often prefer taking the time derivative of the total energy equation (=0) so you'd end up with a homogenous characteristic equation with can usually be solved with the first technique above (plugging in $e^{rt}$ as a solution). This is a very powerful tool for solving complex vibration problems that have, for instance, rotational and linear motion... a bit overkill for simple harmonic motion - but i understand the desire of wanting to find solutions to new problems using old tricks.