Differential equations Definition and 166 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. 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...
  2. 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...
  3. 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...
  4. 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...
  5. 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...
  6. 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...
  7. 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...
  8. 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...
  9. 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...
  10. 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: $$ \begin{aligned} y_1 &= M(x, a, b) \\ y_2 &= x^{1-b}M(x, a-b+1, 2-b) \\ \end{aligned} $$...
  11. yucheng

    Simmons 7.11 Family of curves of a differential equation that intersect 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...
  12. yucheng

    Applied Is Piaggio's Differential Equations worth reading?

    I got to know of this book through Freeman Dyson's obituary. Just wondering, is it useful in studying Physics (it seems to cover everything), do people even use it these days? I understand differential equations are basically half of Physics. By the way, this book is really old, are there any...
  13. R

    I Is it possible to solve such a differential equation?

    Hello, I would like to is it possible to solve such a differential equation (I would like to know the z(x) function): \displaystyle{ \frac{z}{z+dz}= \frac{(x+dx)d(x+dx)}{xdx}} I separated variables z,x to integrate it some way. Then I would get this z(x) function. My idea is to find such...
  14. WyattKEllis

    Diff. Eq. — Identifying Particular Solution Given solution family

    I identified the root 1 with multiplicity 1 and the root 2 with multiplicity 1. So The characteristic equation is ((m-1)^2)*(m-2)=0. Simplifying and substituting with y I found: y'''-4y''+5y'-2y=0. So now I've realized that this is actually describing y(t)=(C1)*(e^t)+(C2)*(e^t)+(C3)*(e^2t) and...
  15. Ron Burgundypants

    Second order differential equation solution

    I know the solution to the equation (1) below can be written in terms of exponential functions or sin and cos as in (2). But I can't remember exactly how to get there using separation of variables. If I separate the quotient on the left and bring a Psi across, aka separation of variables (as I...
  16. J

    Studying Ordinary Differential Equations and Calc III

    Hello, I need help deciding on whether to take ODE (MAP2302) and Calc III during the summer. Would it be wise to take ODE along with Calc III in the same semester? Some people have told me to take Calc III first because there are a few things in ODE that are taught in Calc III, but others have...
  17. L Navarro H

    Find the real functions

    Mod note: Member warned that some effort must be shown.
  18. A

    A Predicting how far an object will fly

    Hi, I have an experimental setup where we are taking certain different types of metals of varying shapes and sizes, weighing them, taking approximate measurements, and then blowing it off of a table of a fixed height with an Air Nozzle. The data taken down in experiment is the PSI at which the...
  19. A

    Differential Equations and Damper Curves

    Good evening, I have been wrestling with the following and thought I would ask for help. I am trying to come up with the equations of motion and energy stored in individual suspension components when a wheel is fired towards the car but, there is a twist! I am assuming a quarter car type...
  20. J

    Heating a soup (solving this problem with a DE)

    I'm having quite a bit of a problem with this one. I've managed to figure out that ##T_0 = 0##. However, not knowing what ##q(t)## is bothers me, although it seems that I could theoretically solve the problem without knowing it. For ##t>t_1##, integration by parts gives me ##T = Ce^{-t/10}##...
  21. L

    Green's functions (Fourier Series)

    In order to obtain equation (3), I think I have to do the Fourier transform in the x direction: \begin{equation} \tilde{G}(k,y,x_0,y_0) = \int_{- \infty}^{\infty} G(x,y,x_0,y_0) e^{-i k x} dx \end{equation} So I have: \begin{equation} -k ^2 \tilde{G}(k,y,x_0,y_0) + \frac{\partial^2...
  22. akin-iii

    I How do I derive a PDE for the volume flow rate of a tilting vessel?

    So the other day, I was pouring beer from a can to a mug and I obviously know the flow rate depends on the height of the beer from the bottom of the can (fluid level in the vessel), angle of tilt and I think time as well. I was wondering how to best model the PDE to describe such a phenomenon (...
  23. S

    Set up the differential equation showing the voltage V(t) for this RC circuit

    Hi all. I have another exam question that I am not so sure about. I've solved similar problems in textbooks but I have a feeling once again that the correct way to solve this problem is much simpler and eluding me. Especially because my answer to a) is already the solution to c) and d) (I did...
  24. K

    I Find the practical resonance frequencies in a system of linear differential equations

    Hi all, I would like to know what is the equation upon which I can use to determine the practical resonance frequencies in a system of second order, linear differential equations. First some definitions: What I mean by practical resonance frequencies, is the frequencies that a second order...
  25. I

    I What is a symmetric ODE / what does it mean when an ODE is symmetric?

    How can an ODE be symmetric? How would you plot an ODE to show off this property? (i.e. what would be the axes?)
  26. Celso

    Show that the terminal speed of a falling spherical object ...

    To write ##v## as a function of time, I wrote the equation ##m\frac{dv}{dt} = c_{2}v^2 + c_{1}v - mg \implies \frac{mdv}{c_{2}v^2 + c_{1}v - mg} = dt## To solve this, I thought about partial fractions, but several factors of ##-c_{1} \pm \sqrt {c_{1}^2 +4c_{2}*mg}## would appear and they don't...
  27. echomochi

    Finding an implicit solution to this differential equation

    Homework Statement Find an equation that defines IMPLICITLY the parameterized family of solutions y(x) of the differential equation: 5xy dy/dx = x2 + y2 Homework Equations y=ux dy/dx = u+xdu/dx C as a constant of integration The Attempt at a Solution I saw a similar D.E. solved using the y=ux...
  28. C

    Flow of Water into a Bowl with Holes

    I'm facing a problem with that rhyming title up there. The design is thus: a downward-facing, vertical pipe with known constant flow and diameter has water flowing out of it, into a short (15cm-91cm) free fall. At the end of that fall is a bowl of indeterminate depth made of steel with holes...
  29. Hawkingo

    Help in solving an inexact differential equation

    Homework Statement The question is to solve the inexact equation by turning it into exact.the equation is ##( x + y + 4 ) d x + ( - x + y + 6 ) d y = 0## Where "x" and "y" are variable. 2. Homework Equations [/B] 1.(x+y+4)=m and (-x+y+6)=n 2.Integrating Factor =##\frac { 1 } { x ^ { 2 } + y...
  30. G

    Transmission line: leakage current differential equation

    Homework Statement I have a coaxial cable with internal conductor of radius r1 and external conductor of radii r2 and r3. The material of the conductors has a conductivity ##\sigma_1##. Between the conductors there is a imperfect dielectric of conductivity ##\sigma_2##. Consider the...
  31. T

    Find v(t) from Newton's Second Law and Differential Equation

    <Moderator's note: Moved from a technical forum and thus no template.> Is what I have done correct ? I want to find v(t) from Sigma F = m*a. I have gravity force mg pointing downward with positive direction and resistive force R = -b*v^2 pointing upwards with negative direction are acting on a...
  32. W

    The coming revolution in physics education

    Classical physics is difficult because it is based on differential equations, and the differential equations of interest are usually unsolvable. The student must invest a lot of time in learning difficult math, and still can only analyze very simple systems. This difficulty arises in the first...
  33. B

    Solving a second order ODE using reduction of order

    Homework Statement Hi there, I have an assignment which involves using reduction of order to solve for a second solution to an ode (the one attached). However this is a method I am new to, and though I have tried several times, I'm somehow getting something wrong because the LHS and RHS are not...
  34. komarxian

    Differential Equations: Solve the following

    Homework Statement Solve the following differential equations/initial value problems: (cosx) y' + (sinx) y = sin2x Homework Equations I've been attempting to use the trig ID sin2x = 2sinxcosx. I am also trying to solve this problem by using p(x)/P(x) and Q(x) The Attempt at a Solution...
  35. komarxian

    Differential Equations, solve the following: y^(4) - y'' - 2y' +2y = 0

    Homework Statement Solve the following differential equations/initial value problem: y^(4) - y'' - 2y' +2y = 0 Hint: e^-x sinx is a solution Homework Equations I was attempting to solve this problem by using a characteristic equation. The Attempt at a Solution y'''' -y'' -2y' + 2y = 0 -->...
  36. C

    Non-linear second-order ODE to Fuchsian equation

    Homework Statement z\frac{d^2z}{dw^2}+\left(\frac{dz}{dw}\right)^2+\frac{\left(2w^2-1\right)}{w^3}z\frac{dz}{dw}+\frac{z^2}{2w^4}=0 (a) Use z=\sqrt y to linearize the equation. (b) Use t=\frac{1}{w} to make singularities regular. (c) Solve the equation. (d) Is the last equation obtained a...
  37. Riotto

    I Can all differential equations be turned into algebraic equations via the FT?

    Can all differential equations be turned into algebraic equations by Fourier transform (FT)? If not, what kind of differential equations can be solved by the FT technique?
  38. I

    I Data Model of Kepler's Second Law of Planetary Motion

    Hello, I am completing a research project for differential equations class. I am to derive Kepler's three laws and then compare the results of the derivation with real-world data. For Kepler's second law (a planet sweeps out an equal area in an equal time), I was hoping to find orbital data for...
  39. Guy Fieri

    Issue With Optimization Problem

    Homework Statement Homework Equations I have yet to figure out any relevant equations, but I do believe that the constraint equation for the optimization problem is the y=64-x^6 listed above. The Attempt at a Solution I am currently trying to figure out methods to begin my optimization...
  40. Jayalk97

    Quick question about linearization using the small angle method

    Hey guys, when you're linearizing a function that has a constant, what do you do to it? An example would be y = x^2 + 3, would you just linearize it using its derivative and get rid of the constant?
  41. Matt Chu

    Proving a complex wave satisfies Helmholtz equation

    Homework Statement Consider a harmonic wave given by $$\Psi (x, t) = U(x, y, z) e^{-i \omega t}$$ where ##U(x, y, z)## is called the complex amplitude. Show that ##U## satisfies the Helmholtz equation: $$ (\nabla + k^2) U (x, y, z) = 0 $$ Homework Equations Everything important already in...
  42. Matt Chu

    Proving a wave satisfies the Helmholtz equation

    Homework Statement Consider a harmonic wave given by $$\Psi (x, t) = U(x, y, z) e^{-i \omega t}$$ where ##U(x, y, z)## is called the complex amplitude. Show that ##U## satisfies the Helmholtz equation: $$ (\nabla + k^2) U (x, y, z) = 0 $$ Homework Equations Everything important already in...
  43. F

    Discharge in a DC RC circuit and Kirchhoff's Loop Rule

    Hi all, I think this issue periodically resurfaces in PF. I have found a similar discussion in this closed post and possibly others. I'm posting this because I'd like to check my understanding, if anyone is available to provide some furtherinsight. So I'm trying to gather a "overall"...
  44. DaxInvader

    Solving 2nd order DE with initial condition

    Hello Guys, We haven't yet covered on how to solve 2nd order equation in class however we have this assignment given to us. Any tips would be appreciated for these 2 little problems. 1. Homework Statement We have this initial Equation: d2y/dt2−7dy/dt+ky=0, and we need to find the values of k...
  45. Spoti112

    Kinematics problem with differential equations.

    Homework Statement Suppose the acceleration of a particle is a function of x, where a(x)=(2.0 s-2)*x. (a) If the velocity is zero when x= 1.0 m, what is the speed when x=3.0 m? (b) How long does it take the particle to travel from x=1.0 m to x=3.0 m. a(x)=(2.0 s-2) * x (a) V(x=3) = ? ...
  46. thepikminman

    Vibrations - Modeling system, equation of motion

    Vibrations - Modelling system, equation of motion Hi, In the first question (question 4) in the attached file, how would you go about modelling the system and finding the equation of motion? All those masses are confusing me, I don't even know where to start. I don't know whether the angle...
  47. J6204

    Calculating the Fourier integral representation of f(x)

    Homework Statement Considering the function $$f(x) = e^{-x}, x>0$$ and $$f(-x) = f(x)$$. I am trying to find the Fourier integral representation of f(x). Homework Equations $$f(x) = \int_0^\infty \left( A(\alpha)\cos\alpha x +B(\alpha) \sin\alpha x\right) d\alpha$$ $$A(\alpha) =...
  48. Ron Burgundypants

    Eigenvalues and vectors of a 4 by 4 matrix

    Homework Statement Coupled Harmonic Oscillators. In this series of exercises you are asked to generalize the material on harmonic oscillators in Section 6.2 to the case where the oscillators are coupled. Suppose there are two masses m1 and m2 attached to springs and walls as shown in Figure...
  49. R

    Coupled differential equations using matrix exponent

    Homework Statement Solve the following coupled differential equations by finding the eigenvectors and eigenvalues of the matrix and using it to calculate the matrix exponent: $$\frac{df}{dz}=i\delta f(z)+i\kappa b(z)$$ $$\frac{db}{dz}=-i\delta b(z)-i\kappa f(z)$$ In matrix form...
  50. A

    I Difference between transient and steady state solution

    In driven SHM, we ignore an entire section of the solution to the differential equation claiming that it disappears once the system reaches a steady state. Can someone elaborate on this?
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