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. Steven Reichman

    Studying Why do I keep failing this in particular? (Differential Equations)

    Hello everyone. I'm an undergrad physics major with one semester left and I'm having some trouble. I took off 3 years to work on my depression and came back last spring to finish my senior year. Now, before I left I was struggling in all my classes due to my depression, but one was worse...
  2. Pouyan

    Fourier series and differential equations

    Homework Statement Find the values of the constant a for which the problem y''(t)+ay(t)=y(t+π), t∈ℝ, has a solution with period 2π which is not identically zero. Also determine all such solutions Homework Equations With help of Fourier series I know that : Cn(y''(t))= -n2*Cn(y(t)) Cn(y(t+π)) =...
  3. P

    Matlab code problem with differential equations

    Homework Statement For a following differential equation d^2y/dx^2-4y=(e^x)/x  Find the solution using numerical methods Homework Equations d^2y/dx^2-4y=(e^x)/x The Attempt at a Solution %num dx=0.01; x=1:dx:3; l=zeros(1,length(x)); m=zeros(1,length(x)); l(1)=1; m(1)=0.25; for...
  4. A

    MHB 2 Differential Equations by Substitution

    solve the following differential equation with the suggested change of variables.
  5. S

    B How were differential equations for SIR Models calculated?

    Specifically:
  6. Cocoleia

    Initial value problem - differential equations

    Homework Statement I am given (y^2 + y sin x cos y) dx + (xy + y cos x sin y) dy = 0, y(0) = π/2 . I need to solve this Homework EquationsThe Attempt at a Solution At this point they still aren't exact, so I gave up. I can't figure out what the problem is. Is it possible that I have to...
  7. Elvis 123456789

    Courses Partial Differential Equations vs Classical Mechanics 2?

    Hello everyone. So I wanted to get some opinions on what some of you thought was a better choice, as far taking PDE's or classical mechanics 2 goes. First let me start off by giving a little info; I've already taken calc 1-3 and ordinary differential equations, physics 1 & 2...
  8. J

    Studying Differential equations with complex functions?

    Hi folks, When you have a differential equation and the unknown function is complex, like in the Schrodinger equation, What methods should you use to solve it? I mean, there is a theory of complex functions, Laurent series, Cauchy integrals and so on, I guess if it would be possible to...
  9. Tspirit

    I How to solve the two following differential equations?

    (1) ##\frac{d^{2}y}{dx^{2}}=0## (2) ##\frac{d^{2}y}{dx^{2}}=k^{2}y##, where k is a real positive number.
  10. E

    Engineering Solving RLC circuit using differential equations

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  11. E

    Engineering Solve OpAmp Circuit Using Differential Equations

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  12. Q

    Identifying Types of Singularity in Differential Equations

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  13. ShayanJ

    A Differential equations without Green functions

    Are there differential equations that, for some reason, don't have a Green function? Are there conditions for a DE to satisfy so that it can have a Green function? Thanks
  14. H

    MHB Differential equations stability

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  15. TheSodesa

    A real parameter guaranteeing subspace invariance

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  16. M

    Physicists' knowledge of differential equations

    <<Moderator's note: this is a spin-off of https://www.physicsforums.com/threads/how-long-to-learn-physics.891250/>> To Zapper. Some differential equations have no analytical solutions but some do. I remember a case where a co worker was using a computer to find the solution to x dot = 1 /...
  17. H

    Arbitrary constant in denominator

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  18. DarkBabylon

    Solving differential equations using numeric methods

    Hello, I have been working on a little movement system in a program called Game Maker: studio. The code works fine on the programming perspective, but something I did not expect happened: When I ran the code by adding to the speed while pressing a key, and every step passively subtracting from...
  19. W

    Linear Differential Equations and Linear Operator Problem

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  20. Elvis 123456789

    How can I be sure of my numerical result?

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  21. Pouyan

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  22. Phylosopher

    Courses Differential equations theory course, is it useful?

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  23. T

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  24. J

    Help finding general solution for 2nd order linear DE

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  25. J

    I What's the geometric interpretation of the trace of a matrix

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  26. jdawg

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  27. C

    Establishing equations for a worm screw mechanism

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  28. tomwilliam2

    I Solving linear differential equations

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  29. RoboNerd

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  30. F

    Ambiguity in the method applied for differential equations

    Homework Statement Why do we need two solutions to solve a 2nd order linear differential equation? lets consider a differential equation with equal roots for auxiliary equation. So the reasoning behind why can't we use y=Aen1x+Ben2x as its general solution is because since the roots are equal...
  31. J

    A Free damped vibration of a system of 2 dof, demostration

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  32. awholenumber

    Calculus First order differential equations ?

    what is a good book to learn first order differential equations ??
  33. awholenumber

    I First order differential equations ?

    what is a good book to learn first order differential equations ??
  34. awholenumber

    I Differential equations and numerical methods questions

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  35. Houeto

    Is this ODE a Bernoulli Equation? Exploring Solutions with Substitution

    consider ODE : Show that the solution to this ODE is: Can someone tell what kind of ODE is it?I thought,it's on the form of Bernoulli ODE with P(x)=0.Is it possible to still solve it by using Bernoulli Methodology?I mean by substituting u=y^1-a with a=2? Thanks
  36. F

    I Diagonalising a system of differential equations

    Given a system of linear differential equations $$x_{1}'=a_{11}x_{1}+a_{12}x_{2}+\cdots a_{1n}x_{n}\\ x_{2}'=a_{21}x_{1}+a_{22}x_{2}+\cdots a_{2n}x_{n}\\ \ldots\\ x_{n}'= a_{n1}x_{1}+a_{n2}x_{2}+\cdots a_{nn}x_{n}$$ this can be rewritten in the form of a matrix equation...
  37. dumbdumNotSmart

    B System of differential equations Basic question

    So I ran into an case I have not seen before. Say we have a system of 3 equations such that W´=AW, where W=(x(t),y(t),z(t)) and A is a 3x3 matrix. The way I usually approach these is by finding the eigenvalues of A to then find the eigenvectors and thus find the ¨homogenous¨ solution. What...
  38. B

    I Laplace transform using differential equations

    Hi members, Laplace transform using differential equations.(see attached PDF file) My question d/ds(s^2 y- s Y(0)-Y'(0).)... Y(t)=sin(sqrt(t)) Y(o)=0 Now Y'= cos(sqrt(t)/2sqrt(t) Y'(0)=infinity d/ds (Y'(0)=?? can it be treated as a constant or can we change limit and differentiation??I...
  39. J

    I William skinner Machine gravity

    Could someone explain under which principle does Skinner's machine operate? Is it just a free falling mass that's taken back to it initial position by very little input energy, thus generating a constant gradient or potential difference, and generating energy, or is it just storing kinetic energy?
  40. J

    An application of free fall (DE) model to industry

    Could someone tell me an application of the model of free fall to industry or more generally by using Newton's second law and the law of gravitation, construct a model similar to the free fall one, that has an application in industry
  41. awholenumber

    B Few basic questions about differential equations....

    i have a few questions to ask about differential equations ... how many types of differential equations are there ... ? sometimes i like to make up themes for my studies ... few funny things went through my head ...when i saw this thread ,How is it that mathematics describe reality so well? i...
  42. H

    I How to find the inverse of an integral transform?

    I'm trying to find the distribution of a random variable ##T## supported on ##[t_1, t_2]## subject to ## \mathbb{E}[V(t', T)] = K, \forall t' \in [t_1, t_2]##. In integral form, this is : $$ \int_{t_1}^{t_2} V(t', t).f(t) \, dt = K,\forall t' \in [t_1, t_2], $$ which is just an exotic integral...
  43. Dusty912

    Convolutions (differential Equations)

    Homework Statement Compute the convolution f*g for the given function f and g. f(t)=cost g(t)=U2(t)Homework Equations f*g=∫f(t-u)g(u)du The Attempt at a Solution So I pretty much only know how to plug in the functions into the integral for convolutions. Not really sure how to evaluate it...
  44. chwala

    Finding Stationary Points for the Differential Equation y=(lnx)^2/x

    Homework Statement Get the two stationary points for the equation ## y= ((ln x)^2)/x ## Homework EquationsThe Attempt at a Solution i have managed to solve ##dy/dx=((2xlnx/x- (ln x)^2))/x^2 = 0, ln x(2-ln x) = 0, x= 1, x =e^2##
  45. J

    Is there an encyclopedia of DE that accepts solutions?

    DE= Differential equations. There is an Encyclopedia of Integer Sequences for example, but I am not able to find the equivalent for DE. I would like to find a list with all the differential equations that have been solved up to date. A website or any other source would be interesting. If the...
  46. R

    Video lectures for differential equations 2

    I took differential equation 2 last semester and the book we used wasn't so great at explaining a lot of things. I was wondering if anyone knew of a video lecture series that parallels my book (Differential Equations 4th Ed. by Blanchard, Devaney, and Hall). Specifically I am having trouble with...
  47. A

    B Complex Replacement: Justification?

    Hi, Is there a proof that complex replacement is a valid way to solve a differential equation? I'm lacking some intuition on the idea that under any algebraic manipulations the real and imaginary parts of an expression don't influence each other. For example, if I'm given: $$p(D) x = cos(t)$$...
  48. faradayscat

    Solve System with Repeated Eigenvalues

    Homework Statement I want to solve this systemx' = \left( \begin{array}\\ 7 & 1 \\ -4 & 3 \end{array} \right)x + \left( \begin{array}\\ t \\ 2t \end{array} \right) Homework EquationsThe Attempt at a Solution i found the eigenvalues to both be 5. The eigenvector is (1,-2) and the generalized...
  49. faradayscat

    I Non-homogeneous systems with repeated eigenvalues

    Quick question, can you solve non-homogeneous systems with repeated eigenvalues the same ways? i.e. variation of parameters, undetermined coefficients, etc... would the fundamental matrix contain the solution with the generalized eigenvalue? Thanks!
  50. T

    How to prepare for differential equations?

    I'm pretty rusty at calculus. I did well in them, but my memory is terrible and I have forgotten a lot. I'm going to take ordinary differential equations (it looks and sounds like an intro DE class with some linear algebra too) next spring. What should I study and what not to prepare for this...
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