Differential equations Definition and 999 Threads

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. T

    Engineering Multi-point Boundary Value Problems Using Finite Difference Method

    Hi all, I am learning how to solve differential equtions using the finite diference method. In particular, for beams under a uniformly distributed load. For a simply supported beam this is quite easy. The boundary conditions are that at each end the displacement equal zero, and using the fourth...
  2. Z

    Why does this method of solving a system of 2nd order differential equations not work?

    I have a question about solving this system. I (naively, I think) initially did the following Trial solution: ##\vec{x}=e^{\lambda t}\vec{a}##. Sub this into the system (2) $$\lambda^2 e^{\lambda t}\vec{a}=M^{-1}Ke^{\lambda t}\vec{a}\tag{3}$$ $$(M^{-1}K-\lambda^2I)\vec{a}=0\tag{4}$$ I then...
  3. Z

    Force transmitted to a block in viscous fluid through a spring, cord, and pulley

    The equation of motions looks like $$m\ddot{x}(t)+m\Gamma\dot{x}(t)=-K(x(t)-d_0\cos{\omega_d t})\tag{3}$$ Moving other end of the spring sinusoidally effectively produces a sinusoidally varying force on the mass. Everything written above so far is as presented by the book "The Physics of...
  4. Mardonio

    Enthalpy derivation differential equation

    Good evening, unfortunately I'm pretty lost in this problem. I tried to use the chain rule $$(\frac {\partial H} {\partial v})_P = (\frac {\partial H} {\partial T})_P (\frac {\partial T} {\partial v})_P$$ and using some Maxwell relations but it doesn't work very well. I know that $$T = (\frac...
  5. Z

    Why can't a critical point of a system of DEs be complex?

    From the first equation we can write $$y=\frac{x}{2}+\frac{x^2}{8}$$ Subbing into the rhs of the second equation and equating to zero we find (after some algebra) that $$x(x-4)(x^2+12x+72)=0$$ This equation has roots ##0##, ##4##, and ##-6\pm 6i##. Then, ##x=0\implies y=0## and...
  6. Z

    How to draw phase portrait for 2x2 nonlinear system of DE?

    The critical points are ##(0,0)## and ##(2,1)##. The linearization of these equations is $$\begin{bmatrix}x'\\y'\end{bmatrix}=\begin{bmatrix}-1+y_0&x_0\\y_0&x_0-2\end{bmatrix}\begin{bmatrix}x-x_0\\y-y_0\end{bmatrix}$$ At ##(0,0)## we have $$\begin{bmatrix}x'\\...
  7. Z

    Relationship between autonomous system and related single equation

    Here are the notes. We have the system $$\begin{bmatrix} x'\\y' \end{bmatrix}=\begin{bmatrix}f(x,y)\\g(x,y)\end{bmatrix}\tag{1}$$ We eliminate ##t## by dividing one equation by the other $$\frac{y'}{x'}=\frac{dy/dt}{dx/dt}=\frac{dy}{dx}=\frac{g(x,y)}{f(x,y)}\tag{2}$$...
  8. cianfa72

    I Are Maxwell's equations linearly dependent?

    HI, consider the 4 Maxwell's equations in microscopic/vacuum formulation as for example described here Maxwell's equations (in the following one assumes charge density ##\rho## and current density ##J## as assigned -- i.e. they are not unknowns but are given as functions of space and time...
  9. M

    A Decoupling of a spring mass system

    Hi guys Please refer to the attached image. It is really easy to derive a set of differential equations which present a spring on a mass system, however how can one consider a system where the mass and spring can decouple? The first image on the left shows a spring a rest with a mass which...
  10. Z

    Verify Green's Formula for a Simple DE

    These problems are from a practice problem set from MIT OCW's 18.03 "Differential Equations. Computing the convolutions ##t*1## and ##1*t## is straightforward. They both equal ##\frac{t^2}{2}##. Then, ##(q*1)(t)=\int_0^t q(\tau)d\tau## and ##(1*q)(t)=\int_0^t q(t-\tau)d\tau## which after a...
  11. C

    Solve (t^2-1)y'' +4ty'+2y=6t, given two particular solutions

    I find a solution in math.exchange site: https://math.stackexchange.com/questions/3100237/find-general-solution-given-two-particular-solutions The way I thought about solving the problem is to somehow use the two particular solutions to generate a homogeneous solution, I couldn't figure out how...
  12. C

    Solving differential equation (t^2-1)y''-6y=1

    This was a question from some past exam, I found online a solution but it uses Frobenius method which wasn't taught in the course. I would approach the solution by attempting to find a solution to the homogeneuous DE ##\left(t^2-1\right) \ddot{y}-6 y=0##, but that by itself is quite tricky...
  13. Z

    Given unit impulse response, obtain differential equation

    I initially solved this problem in quite a roundabout way by thinking about a mass-spring-dashpot system modeled by $$m\ddot{x}+b\dot{x}+kx=f(t)$$ Since the response is constant at ##x=0## and has a jump at ##t=0## to ##x=1##, my reasoning was that there can be no spring otherwise there would...
  14. Z

    Obtaining stability criterion for 2nd order ODEs in coefficient form

    As we noted above, stability is all about the solution to the homogeneous equation. For the equation $$y''+by'+ay=0\tag{3}$$ we have discriminant $$\Delta = b^2-4a\tag{4}$$ and the roots are $$r=\frac{-b\pm\sqrt{b^2-4a}}{2}\tag{5}$$ We have three cases. Case 1 (Distinct Real Non-Complex...
  15. Z

    What Makes the Undetermined Coefficients Method Fail with Sinusoidal Input?

    If ##p^2-4q<0## then we know that the homogeneous equation has a general solution $$y_g(x)=c_1\sin{kx}+c_2\cos{kx}\tag{3}$$ where $$k=\frac{1}{2}\sqrt{-\Delta}=\frac{\sqrt{4q-p^2}}{2}\tag{4}$$ Suppose we guess at a solution ##y_p## to the non-homogeneous equation...
  16. Z

    Linearly independent functions with identically zero Wronskian

    My question will be about item (c). Part (a) Note that for ##x\geq 0## we have ##f(x)=g(x)##. For ##x<0## we have ##f(x)=-g(x)##. Since ##f## is a constant times ##g## then one column of the matrix in the Wronskian is a constant times the other column. Thus, the Wronskian is zero, Note that...
  17. Z

    Why is this not a general solution to this nonlinear DE?

    a) We can use reduction of order $$p=y'\tag{1}$$ $$p'=y''\tag{2}$$ The DE becomes $$p'+p^2=0\tag{3}$$ $$\frac{1}{p^2}p'=-1\tag{4}$$ This last step contains the assumption that ##p^2=y'^2\neq 0##. $$-\left (\frac{1}{p(x)}-\frac{1}{p(x_i)}\right )=-(x-x_i)\tag{5}$$...
  18. Z

    Understanding Wronskian of solutions to homoeneous 2nd order linear DE

    The Wronskian of these two solutions is also a function of ##x##. $$W=y_1y_2'-y_1'y_2$$ $$W'=y_1y_2''+y_1'y_2'-y_1'y_2'-y_1''y_2$$ $$=y_1y_2''-y_1''y_2$$ The two solutions satisfy $$y_1''+Py_1'+Qy_1=0$$ $$y_2''+Py_2'+Qy_2=0$$ Multiply the first by ##y_2## and the second by ##y_1## and...
  19. chwala

    I Determining linear indepedence/dependence of a set of functions

    Looking at the wronskian applications- came across this; Okay, i noted that one can also have this approach(just differentiate directly). Sharing just incase one has more insight. ##-18c \sin 2x -4k\cos x \sin x - 4k\sin x\cos x =0## ##-18c\sin 2x-2k\sin2x-2k\sin 2x=0## ##-18c\sin 2x =...
  20. Davidian

    Variation of parameter VS Undetermined Coefficients

    TL;DR Summary: Variation of parameter VS Undetermined Coefficients Hi all, Suppose we want to solve the following ODE 2y''+y'-y=x+7 with two different methods: undetermined coefficients and variation of parameters. The solutions to the homogeneous problem are given by y_1(x)=exp(-x) and...
  21. berlinvic

    Solving system of differential equations using elimination method

    I am trying to solve this system of differential equations using elimination method, but I am stuck. $$\begin{cases} y'_1 = y_2, \\ y'_2 = -y_1 + \frac{1}{\cos x} \end{cases}$$ Here's what I tried: I've been suggested to differentiate the ##y_1'= y_2## again to get ##y_1''= y_2'=...
  22. Z

    Obtain a nonzero solution of ##y''-4y'+x^2(y'-4y)=0## by inspection

    I have not been able to solve this. Here is what I tried to do. ##z=y'-4y## ##z'=y''-4y'## Thus, the second order equation in ##y## becomes ##z'+x^2z=0##, a first order equation in ##z##, the solution to which is ##z(x)=ke^{-\frac{x^3}{3}}## with ##k>0##. Thus...
  23. 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, $$\left[\int -\dfrac{1}{y} dy - \int \dfrac{1}{y-1} dy\right] = \int \dfrac {1}{6} dt$$...
  24. 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...
  25. 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. $$\frac{dv}{dt}=\frac{dv}{ds}\frac{ds}{dt}=v\frac{dv}{ds}$$ $$\frac{ds}{dv}=\frac{v}{v'}=\frac{v}{ge^{-kt/m}}$$ $$=\frac{\frac{m}{k}v}{\frac{gm}{k}e^{-kt/m}}$$...
  26. 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...
  27. 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...
  28. 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...
  29. 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 a...
  30. 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...
  31. S

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

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

    A How to Solve This Differential Equation Analytically?

    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
  33. 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...
  34. 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...
  35. 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?
  36. 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...
  37. 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...
  38. 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...
  39. 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...
  40. 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...
  41. 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...
  42. 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.
  43. M

    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...
  44. 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: $${5} \frac{\text{d}\theta}{\text{d}t}=-6\theta$$ $${5} \frac{\text{d}\theta}{\text{d}t} =\frac{\text{-6}\theta}{5}$$...
  45. 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.
  46. 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...
  47. 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...
  48. 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...
  49. 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...
  50. 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...
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