What is Differential equations: Definition and 998 Discussions

In mathematics, a differential equation is an equation that relates one or more functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.
Mainly the study of differential equations consists of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly.
Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.

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  1. chwala

    I Determining equilibrium solutions to differential equations

    I am going through this, I noted that, i shall have a separation of variables, that leads to $$\left[\int \dfrac{1}{y(y-1)} dy\right]= \int \dfrac {1}{6} dt$$ and using partial fraction, i then have, $$\left[\int -\dfrac{1}{y} dy - \int \dfrac{1}{y-1} dy\right] = \int \dfrac {1}{6} dt$$...
  2. Z

    Solve ##\int\frac{e^{-x}}{x^2}dx## and ##\int \frac{e^{-x}}{x}dx##

    The characteristic equation has a zero discriminant and the sole root of ##-1##. The general solution to the associated homogeneous equation is thus $$y_h(x)=e^{-x}(c_1+c_2x)\tag{1}$$ Now we only need to find one particular solution of the non-homogeneous equation. The righthand side of the...
  3. Z

    Apostol question about the differential equations of a falling object

    Here is my solution to this problem. Unfortunately, I can't check it because it is not contained in the solution manual. $$\frac{dv}{dt}=\frac{dv}{ds}\frac{ds}{dt}=v\frac{dv}{ds}$$ $$\frac{ds}{dv}=\frac{v}{v'}=\frac{v}{ge^{-kt/m}}$$ $$=\frac{\frac{m}{k}v}{\frac{gm}{k}e^{-kt/m}}$$...
  4. M

    Calculus ## G(y, z)=z\frac{\partial}{\partial z}F(y, z)-F(y, z) ##?

    a) Observe that ## \frac{\partial}{\partial z}F(y, z)=y^{n-1}\cdot \frac{2z}{2\sqrt{y^2+z^2}}=\frac{zy^{n-1}}{\sqrt{y^2+z^2}} ##. This means ## G(y, z)=\frac{z^2\cdot y^{n-1}}{\sqrt{y^2+z^2}}-y^{n-1}\cdot \sqrt{y^2+z^2}=\frac{z^2\cdot...
  5. P

    I Feynmann's connection machine model

    Lowly engineer here. I am struggling - I think like many - to develop intuition on DEs. From looking at the history and applications of DEs, general themes that come to mind are, conservation, energy (eg. isochrone problem), causality, feedback (control systems), etc. However, I can't seem to...
  6. T

    A F.E.M and Hamilton's Principle (converting differential equations into integral equations)

    Hello May I begin by saying I do not exactly know what I am asking, but here goes... In the Finite Element Method (as used in Solid Mechanics), we convert the differential equations of continuum mechanics into integral form. Here, I am thinking of the more pragmatic Principle of Virtual Work...
  7. L

    Help using Green’s functions in solving Differential Equations please

    Hi, unfortunately I have several problems with the following task: I have problems with the tasks a, d and e Unfortunately, the Green function and solving differential equations with the Green function is completely new to me In task b, I got the following for ##f_h(t)=e^{-at}##. Task...
  8. T

    A Separation of variables is possible only in 11 coordinate systems?

    I vaguely (strong word there because I can no longer remember the source, but the idea sticks in my head for 30 years now) recall reading (somewhere long forgotten) that method of separation of variables is possible in only 11 coordinate systems. I list them below: 1.Cartesian coordinates...
  9. S

    I What's the mathematics behind "spontaneous synchronization"?

    I was looking at this video, and have become quite interested:
  10. S

    A How to integrate this equation?

    Hello, can someone help me to solve the following differential equation analitically: $$\frac{2 y''}{y'} - \frac{y'}{y} = \frac{x'}{x}$$ where ##y = y(t)## and ##x = x(t)## br Santiago
  11. matqkks

    What to cover in a differential equations module?

    I will have to teach a first course in differential equations. What should I cover in this module? For example, in most books they have Laplace Transforms which is fine but I would not use LT to solve differential equations. I want to write a course that it motivates students and has an impact...
  12. D

    A Heat exchange in a thermal storage based on phase change materials

    Hello, I want to model a thermal battery based on phase change materials (PCM). It is a plate heat exchanger immersed in a PCM bath. The diagram is given in the attached file. I want to determine the temperature at each moment and from everywhere in the battery. The hypotheses are the...
  13. L

    A Applying the Laplace transform to solve Differential equations

    Is it possible to apply Laplace transform to some equation of finite order, second for instance, and get the differential equation of infinite order?
  14. H

    Prof. Mattuck's Lecture 20: A Divine Power of Speech & Laplace's Elegance

    Lecture 20 by Prof. Mattuck is simply great. I don’t think there can be any better communication than that, it seemed as if what was in his mind simply got teleported in to mine, and that’s why, it seems now, Aristotle gave so much importance (and almost considered it as a divine power) to the...
  15. H

    Calculus Following Prof. Mattuck's lectures on Ordinary Differential Equations....

    I'm learning Differential Equations from Prof. Mattuck's lectures. The lectures are absolutely incredible. But there are a few topics in Tenenbaum's book and my syllabus which he doesn't seem to teach (I have reached upto lecture 14, but in future lectures too the following topics are not...
  16. T

    Integration of acceleration in polar coordinates

    I made this exercise up to acquire more skill with polar coordinates. The idea is you're given the acceleration vector and have to find the position vector corresponding to it, working in reverse of the image. My attempts are the following, I proceed using 3 "independent" methods just as you...
  17. H

    Calculus Differential Equations book recommendations

    I ordered Differential Equations and Boundary Value Problem ( Computing and Modelling) by Edwards and Penney. There are several things in the book which I don't like Too much focus is given to modelling, almost every topic is explained not from mathematical point of view but from application...
  18. U

    Looking for Friend to Discuss Differential Equations

    [Mentor Note -- This thread start is by a new member from the recent MHB forum merger] Hello Guys, I want to find a friend with whom I can discuss differential equations! I would like to do that via WhatsApp or zoom applications. I am interested in applications of differential equations ( for...
  19. Haorong Wu

    Solutions of first-order matrix differential equations

    Hello, there. I am trying to solve the differential equation, ##[A(t)+B(t) \partial_t]\left | \psi \right >=0 ##. However, ##A(t)## and ##B(t)## can not be simultaneous diagonalized. I do not know is there any method that can apprixmately solve the equation. I suppose I could write the...
  20. E

    MHB Oil Spill - Differential Equations

    Hello everyone. I hope anyone can help me with this problem. I will greatly appreciate it. Willing to compensate anybody to answer this problem correctly for me.
  21. Vick

    A Peebles Equation for fractional electron density

    I am trying to compute the Peebles equation as found here: I am doing so in Python and the following is my attempt: However, I'm unable to solve it. Either my solver is not enough, or I have wrongly done the function for calculating the Equation. # imports from scipy.optimize import fsolve...
  22. B

    Engineering First order differential equations (movement of a rotary solenoid)

    My question i am trying to solve: I have successfully done first order equations before but this one has got me a little stuck. My attempt at the general solution below: $${5} \frac{\text{d}\theta}{\text{d}t}=-6\theta$$ $${5} \frac{\text{d}\theta}{\text{d}t} =\frac{\text{-6}\theta}{5}$$...
  23. chwala

    Proof involving ##ω(ξ,n)=u(x,y)## - Partial differential equations

    I am going through this page again...just out of curiosity, how did they arrive at the given transforms?, ...i think i get it...very confusing... in general, ##U_{xx} = ξ_{xx} =ξ_{x}ξ_{x}= ξ^2_{x}## . Also we may have ##U_{xy} =ξ_{xy} =ξ_{x}ξ_{y}.## the other transforms follow in a similar manner.
  24. M

    From differential equations to transfer functions

    *** MENTOR NOTE: This thread was moved from another forum to this forum hence no homework template. Summary:: Trying to find transfer functions to design a block diagram on simulink with a PID controller and transfer functions for a water tank system. ----EDIT--- The variables and parameters...
  25. Maxicl14

    I EM equations - am I missing something?

    Summary:: There seems to be a mismatch, in the "Maxwell's" equations, between the number of equations and number of variables. I was trying to play around with the equations for Electromagnetism and noticed something unusual. When expanded, there are 8 equations, 6 unknown variables, and 4...
  26. V

    Trouble with a Rocket Propulsion question (Variable Mass & Momentum)

    I chose to set the upwards direction to be positive and dM/dt = R = 190 kg/s, so I can solve the problem in variable form and plug in. With the only external force being gravity, this gives M(t) * dv/dt = -M(t) * g + v_rel * R where M(t) is the remaining mass of the rocket. Rearranging this...
  27. The Bill

    Applied Resources for general vector differential equations?

    I'd like a good set of notes or a textbook recommendation on how to approach vector differential equations. I'm looking for something that isn't specific to one type of application like E&M, fluid dynamics, etc., but draws heavily from those and other fields for examples. I'd strongly prefer a...
  28. A

    Domain of definition differential equations

    Hello. Considering this DE; $$ x^7 x' = (x^8-300)t^6 $$ with inital value x(0) = -2 Now the solution for the initial value should be C = -44; And for x(t) I get ; $$x(t) = (-44 e^{\frac{8}{7} t^7} + 300)^{\frac{1}{8}}$$ Now to get the biggest domain of definition I did this; $$ -44...
  29. V

    I Limit cycles, differential equations and Bendixson's criterion

    I have a few questions about the negative Bendixon criterion. In order to present my doubts, I organize this post as follows. First, I present the theorem and its interpretation. Second, I present a worked example and my doubts. The Bendixson criterion is a theorem that permits one to establish...
  30. J

    I Integral-form change of variable in differential equation

    I have the following differential equation, which is the general Sturm-Liouville problem, $$ \dfrac{d}{dx} \left[ p(x) \dfrac{d\varphi}{dx} \right] + \left[ \lambda w(x) - q(x) \right] \varphi(x) = 0\ , $$ and I want to perform the change of variable $$ x \rightarrow y = \int_a^x \sqrt{\lambda...
  31. W

    A Solving nonlinear singular differential equations

    hi, i am going through differential equations which are nonlinear and singular - like Lane-Emden equation etc. my question is how to tackle singularity - while coding. regards
  32. chwala

    Using separation of variables in solving partial differential equations

    I am reading on this part; and i realize that i get confused with the 'lettering' used... i will use my own approach because in that way i am able to work on the pde's at ease and most importantly i understand the concept on separation of variables and therefore would not want to keep on second...
  33. U

    Help with a calculation about gravitational waves

    An exact gravitational plane wave solution to Einstein's field equation has the line metric $$\mathrm{d}s^2=-2\mathrm{d}u\mathrm{d}v+a^2(u)\mathrm{d}^2x+b^2(u)\mathrm{d}^2y.$$ I have calculated the non-vanishing Christoffel symbols and Ricci curvature components and used the vacuum Einstein...
  34. B

    MHB First order differential equations

    Hi, Is the answer: y(x) _homogenous =v(x) y(x) _private =u(x)v(x) ? Or they refer to something else? I don't know how to approach to it
  35. galaxywanderer

    Courses Differential Equations Are Eating Me Alive

    So I am a sophomore physics major at a university near my hometown. I have always been fascinated by the way studying physics makes me think about the world, and I have struggled with but enjoyed my other undergraduate physics and math classes. This semester, however, I am taking multivariable...
  36. A

    Solving a system of differential equations by fundamental matrix

    I am given this system of differential equations; $$ x_1'=2t^2x_1+3t^2x_2+t^5 $$ $$ x_2' =-2t^2x_1-3t^2x_2 +t^2 $$ Now the first question states the following; Find a fundamental matrix of the corresponding homogeneous system and explain exactly how you arrive at independent solutions And the...
  37. M

    Reduction of Order Problem for Differential Equations Class

    Problem statement: Second order linear differential equation in standard from Reasoning:
  38. Tapias5000

    How to solve this application problem in edo?

    Summary:: A nitric acid solution enters at a constant rate of 6 liters / minute into a large tank that originally contained 200 liters of a 0.5% nitric acid solution. The solution inside the tank is kept well stirred and leaves the tank at a rate of 8 liters / minute. If the solution entering...
  39. S

    MHB Solve Differential Eq: xe^-1/(k+e^-1) for x, k, t

    The number of organisms in a population at time t is denoted by x. Treating x as a continuous variable, the differential equation satisfied by x and t is dx/dt= xe^-1/(k+e^-1), where k is a positive constant.. Given that x =10 when t=0 solve the differential equation, obtaining a relation...
  40. L

    I Solve second order linear differential equation

    Consider the second order linear ODE with parameters ##a, b##: $$ xy'' + (b-x)y' - ay = 0 $$ By considering the series solution ##y=\sum c_mx^m##, I have obtained two solutions of the following form: $$ \begin{aligned} y_1 &= M(x, a, b) \\ y_2 &= x^{1-b}M(x, a-b+1, 2-b) \\ \end{aligned} $$...
  41. N

    B Basic Idea of Differential Equations

    Hello. After a lot of researching, I am still not clear how the subject of differential equations is really any different from derivatives and integrals which are learned in the main part of calculus. For example: "Population growth of rabbits: N = the population of rabbits at any time t r=...
  42. chwala

    Exact differential equations

    I am looking at this and i would like some clarity... at the step where "he let" ##μ_y##=0" Could we also use the approach, ##μ_x##=0"? so that we now have, ##μ_y##M=μ(##N_{x} -M_{y})##... and so on, is this also correct?
  43. guyvsdcsniper

    Courses Where to go after Differential Equations?

    I am currently pursuing a Bachelors in Physics. With my current work experience, that degree will eventually allow me to reach an engineering position in Non Destructive Testing. While I enjoy the career field I believe I could do more with my degree. I personally would like to work at LHC or...

    A Phase Portraits of a system of differential equations

    One thing that bothers me regarding the phase portraits, if I plot a phase portrait, then all my possible solutions (for different initial conditions) are included in the diagram? In other words, a phase portrait of a system of ODE's is its characteristic diagram?
  45. greg_rack

    Learning DEs: Solving 2nd Order Differential Equations

    Hi guys, I have just started studying DEs on my own, so pardonne moi in advance for the probably silly question :) Via Newton's second law of motion: $$x''=\frac{F}{m} \ [1]$$ Which is a second-order differential equation. But, from here, how do I get the good old equation of motion...
  46. greg_rack

    B Real world applications of differential equations

    Hi guys, how are you doing? My maths teacher asked me to work on and deliver an engaging insight-oriented "lesson" to my class, about physical/engineering and real-world applications of differential equations, in order to better get the meaning of operating with such mathematical objects. Of...
  47. E

    Studying Need some advice -- Studying oscillations before differential equations?

    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...
  48. Falgun

    Applied Ordinary Differential Equations Books

    I am trying to self study Ordinary Differential Equations and am totally fed up of "cookbook style ODEs". I have recently finished Hubbard's Multivariable Calculus Book and Strang's Linear algebra book. I would like a rigorous and Comprehensive book on ODEs. I have shortlisted a few books below...
  49. yucheng

    Simmons 7.10 & 7.11: Find Curves Intersecting at Angle pi/4

    >10. Let a family of curves be integral curves of a differential equation ##y^{\prime}=f(x, y) .## Let a second family have the property that at each point ##P=(x, y)## the angle from the curve of the first family through ##P## to the curve of the second family through ##P## is ##\alpha .## Show...
  50. V

    Solve the system of differential equations

    I have my set of differential equations which is dx/dt = -2x, dy/dt=-y+x2, with the initial conditions x(0)=x0 and y(0)=y0. I'm a little confused about how to approach this problem. I thought at first I would differentiate both sides of dx/dt = -2x in order to get d2x/dt2 = -2, and then I would...