What is Differential equations: Definition and 999 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. D

    MHB Converting Higher Differential Equations into First Order Systems (Examples and Notes)

    First, some may ask why would do we care that we can convert a 3rd order or higher ODE into a system of equations? Well there are quite a few reasons. 1- Almost all first order systems are easier to solve numerically using computer systems (matlab, maple, etc). Yes, it takes some working out...
  2. 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...
  3. G

    I Symmetry in differential equations

    Before I delve into this , I just wanted to know the basic approach. Do we look for symmetries because it gives us a systematic way to find coordinate changes that change the differential equation into a separable one? Thanks jf
  4. 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...
  5. 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?
  6. A

    I Solving the differential equations involving SHM

    What is the most satisfactory explanation for guessing certain solutions to the differential equations encountered in damped & driven SHM?
  7. L

    If a constant number h of fish are harvested from a fishery

    Hi! Can anyone help me? If a constant number h of fish are harvested from a fishery per unit time, then a model for the population P(t) of the fishery at time t is given by: dP/dt = P(5-P) - h, P(0) = P0. a. Solve for the IVP if h = 4. b. Determine the value of P0 such that the fish...
  8. R

    Maple System of differential equations in Maple

    Hi everybody. I'm using the Maple 13 software (in linux mint) to solve system compounded by the four below differential equations: > ode1 := (diff(m1(t), t)) = - m1(t) + (1/2)*tanh( m2(t) + m4(t) + cos(t) ); > ode2 := (diff(m2(t), t)) = - m2(t) + (1/2)*tanh( m1(t) + cos(t) ); > ode3 :=...
  9. Jianphys17

    I The role of the weight function for adjoint DO

    Hi at all, I've a curiosity about the role that the weight function w(t) she has, into the define of adjoint & s-adjoint op. It is relevant in physical applications or not ?
  10. F

    I Question about second order linear differential equations

    Hi everybody. I need to learn how to solve this kind of equation by decomposing y in a serie of functions. All the examples I have seen are of homogeneous functions. I would be extremely thankfull if someone pointed me to some text in which this is done-explained. Thanks for reading.
  11. W

    Differential Equations: LRC

    Homework Statement How does one show that q(t) is indeed a solution? Homework EquationsThe Attempt at a Solution My current idea is that i should come up with any form of solution, like q = Acos(ωt), and slot it in the RHS. Reason being that if q is indeed a solution, the result of the...
  12. Marcus95

    Coupled differential equations using matrices

    Homework Statement We can treat the following coupled system of differential equations as an eigenvalue problem: ## 2 \frac{dy_1}{dt} = 2f_1 - 3y_1 + y_2 ## ## 2\frac{dy_2}{dt} = 2f_2 + y_1 -3y_2 ## ## \frac{dy_3}{dt} = f_3 - 4y_3 ## where f1, f2 and f3 is a set of time-dependent sources, and...
  13. W

    Differential equations with eigenvalues.

    Homework Statement Find all solutions of the given differential equations: ## \frac{dx}{dt} = \begin{bmatrix} 6 & -3 \\ 2 & 1 \end{bmatrix} x ## Homework EquationsThe Attempt at a Solution So, we just take the determinate of A-I##\lambda## and set it equal to 0 to get the eigenvalues of 3...
  14. Saracen Rue

    Second Order Differential Equations - Beam Deflections

    Homework Statement A cantilever of length ##L## is rigidly fixed at one end and is horizontal in the unstrainted position. If a load is added at the free end of the beam, the downward deflection, ##y##, at a distance, ##x##, along the beam satisfies the differential equation...
  15. Saracen Rue

    B Second derivative differential equations in terms of y?

    Firstly I know how to do this with first derivatives in differential equations - for example say we had ##\frac{dy}{dx}=4y^2-y##, and we're also told that ##y=1## when ##x=0##. ##\frac{dy}{dx}=4y^2-y## ##\frac{dx}{dy}=\frac{1}{4y^2-y}=\frac{1}{y\left(4y-1\right)}=\frac{4}{4y-1}-\frac{1}{y}##...
  16. Rodrigo Schmidt

    Mathematically rigorous Calculus 2 book

    So, i am currently studying physics in a brazilian university. I am going to have a Calculus 2 course which, in Brazil, covers Ordinary Differential Equations and multi-variable differential calculus. So which challenging and rigourous books would you guys recommend for that? Thanks for the...
  17. K

    A A system of partial differential equations with complex vari

    Hi, I need to solve a system of first order partial differential equations with complex variables given by What software should I use for solving this problem..? The system includes 13 differential equations ...
  18. awholenumber

    I What are partial differential equations?

    If the slope of the curve (derivative) at a given point is a number .
  19. Arman777

    Max values of current and charge using differential equations

    Homework Statement Homework Equations Circuit Equations. ##U_C=Q^2\2C## ##U_L=Li^2\2## The Attempt at a Solution For (a) I said ##100J## .But I think it might be ##200J## too.Here what I did; ##U_t=Q^2\2C## and I put ##Q=0.1C## and we know ##C##.Here I...
  20. C

    I Solve Mystery of Phi-Based Equation: Help Colin

    If you know phi it is about 1.618...=2cos36. The equations when x=phi which is equal to 0 is x^2-x-1=0. I took the first derivative squared and the second derivative cubed. The equation with x=phi is: [2x-1]^2+2^3=13 Check for yourself, if you fill in phi you get 13. Anyway, I do not know what...
  21. A

    A Stochastic differential equations with time uncertainty....

    Hi all, I'm wondering if anyone is able to point me in a direction regarding an aspect of stochastic differential equations. I have a situation in which I need to propagate a stochastic DE through time using measurement updates - however, the exact time at which each measurement arrives is...
  22. kupid

    MHB Few beginner doubts about differential equations ?

    I was trying to picture the third derivative of something Then i came across these ... What does displacement mean? The variable x is often used to represent the horizontal position. The variable y is often used to represent the vertical position Displacement=Delta x=xf-x0xf refers to the...
  23. R

    Integral form of Particular solution question

    Homework Statement I'm fine with the first part. Part b) is causing me trouble http://imgur.com/xA9CG5G Homework EquationsThe Attempt at a Solution I tried subbing in the solution y1 into the given equation, but I'm not sure how to differentiate this, i thought of using integration by parts...
  24. gelfand

    Drag force with differential equations, finding max speed

    Homework Statement A submarine engine provides maximum constant force ##F## to propel it through the water. Assume that the magnitude of the resistive drag force of the water experienced by the submarine is ##kv##, where ##k## is the drag coefficient and ##v## is the instantaneous speed of...
  25. F

    I Second order DE with Sine function

    I have this second order differential equation but I'm stumped as to how to solve this since the zeroth order term has a Sine function in it and the variable is embedded. ##\ddot y(t) + 3H (1+Q) \dot y(t) -m^2 f \sin(\frac{y(t)}{f}) = 0## ##H~##, ##~Q~##, ##~m~##, and ##~f~## are just...
  26. patrickmoloney

    What is the solution for calculating population growth with mice?

    Homework Statement Hey guys I'm struggling to find much information of modelling single species population dynamics that relates to this question. A question like this is going to be coming up in my final exam and I need to be able to solve it. I'm struggling to even know where to start. I'm...
  27. cg78ithaca

    A Modeling diffusion and convection in a complex system

    I am trying to come up with an analytical solution (even as a infinite series etc.) for the following diffusion-convection problem. A thin layer of gel (assumed rectangular) is in direct contact with a liquid layer (perfusate) flowing with velocity v in the x direction (left to right) just...
  28. M

    Solution to complex valued ODE

    Homework Statement Let f : I → C be a smooth complex valued function and t0 ∈ I fixed. (i) Show that the initial value problem z'(t) = f(t)z(t) z(t0) = z0 ∈ C has the unique solution z(t) = z0exp(∫f(s)ds) (where the integral runs from t0 to t. Hint : for uniqueness let w(t) be another...
  29. R

    I Parameterization by differential equations

    Hi Is it possible to solve something like this (and are there any errors in the math)? A given curve with implicit function f(x,y) = 0 (for example r^2-x^2-y^2 = 0), has a normal (df/dx, df/dy) and a tangent with direction according to (-df/dy, df/dx). A parameterization of the implicit...
  30. Poetria

    Differential equations (swinging door)

    Homework Statement [/B] There is a swing door with a damper. The characteristic polynomial (I have done it correctly) is: 0.5*r^2+1.5*r+0.625 General solution for x(0)=x_0 and v(0)=v_0 is (I have found it without a problem): (1.25*x_0+v_0/2)*e^(-0.5*t)+((v_0+0.5*x_0)/(-2))*e^(-2.5*t) Now the...
  31. MAGNIBORO

    I Differential Equations For Solving A Recursive equation

    Hi, i have a question about a proof of some recursive equation, the function is $$c_{n}(a)=\int_{0}^{\pi } \frac{cos(nx)-cos(na)}{cos(x)-cos(a)}$$ whit ##n\in \mathbb{N}## and ##a\in \mathbb{R}## . whit some algebra is easy to see ##c_{0}(a)=0## and ##c_{1}(a)=\pi## and the recursive...
  32. W

    I What differential equations need to be solved more quickly?

    For what differential equations would having much quicker or financially cheaper methods of solving them significantly benefit scientists or engineers?
  33. B

    A Test if 2nd order diff eq. can be derived from a Hamiltonian

    Imagine I have a complicated second-order differential equation that I strongly suspect can be derived from a Hamiltonian (with additional momentum dependence beyond p2/2m, so the momentum is not simply mv, but I don't know what it is). Are there any ways to test whether or not the given...
  34. M

    Solve Snow Plow Problem: Find Constant k

    Hi guys. I am currently stuck on the classic snow plow problem. I have the following differential equation and initial conditions: @ 7am plow starts off to clear snow at a constant rate By 8am, plow has gone 4mi By 9am, plow has gone an additional 3mi Let t=0 when it started to snow, when did...
  35. G

    Particle Motion (Astrophysics)

    Homework Statement This is new for me, so forgive me my clumsiness. I am working on the following problem: A particle p is moving with a velocity v1 = c (speed of light) towards an object q, which is moving in the same direction with the speed v2, where v1>v2. Now, v2 is a function of the...
  36. A

    Projectile Motion problem involving air resistance

    Homework Statement Ok, so I am attempting to solve a projectile motion problem involving air resistance that requires me to find the total x-distance the projectile traverses before landing again. Given: \\ m=0.7\text{kg} \\ k=0.01 \frac{\text{kg}}{\text{m}} \\ \theta=30 \degree Homework...
  37. I

    System of differential equations in MATlab simulink

    Hello everyone, I would like if someone could help me with a little excersie here. 1. Homework Statement I am trying to simulate a mechanical system from a differential equations in simulink, but I don't know If I am doing it right. I've made the model as you can see in the 2nd picture...
  38. S

    I Rescaling the equation of motion of inflation

    From the equation of motion of inflation, $$\frac{d^2\phi}{dt^2} + 3H\frac{d\phi}{dt} + \frac{dV}{d\phi} = 0$$ Example: ##V= \frac{1}{2}m^2\phi^2## $$\frac{d^2\phi}{dt^2} + 3H\frac{d\phi}{dt} + m^2\phi = 0$$ If I want to make the DE dimensionless then I let ##~t = \frac{1}{H_o} \tilde t~## and...
  39. J

    Help understanding a vibrating string question

    Homework Statement So I don't really understand what the professor means by "show why the displacements y(x,t) should satisfy this boundary value problem" in problem 1. Doesn't that basically boil down to deriving the wave equation? At least in problem 2 he says what he wants us to show...
  40. Vanessa Avila

    Initial Value Problem for (DE)

    Homework Statement dv/dt = 9.8 - (v/5) , v(0) = 0 (a) The time it must elapse for the objet to reach 98% of its limiting velocity (b) How far does the object fall in the time found in part (a)? Homework Equations (dv/dt)/(9.8-(v/5)) The Attempt at a Solution I'm a little overwhelmed by this...
  41. S

    MATLAB Is MATLAB better for numerical simulation

    From cosmology, the tensor to scalar ratio is ##r=16\epsilon## where ##\epsilon=-\frac{\dot H}{H^2}## is the Hubble slow roll parameter. From warm inflation, $$\ddot \phi + (3H+\Gamma)\dot \phi + V_\phi = 0 ,\quad H^2 = \frac{1}{3M_p^2} (\frac{1}{2} \dot \phi^2 + V)$$ where ##H## is the Hubble...
  42. 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...
  43. 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)) =...
  44. 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...
  45. A

    MHB 2 Differential Equations by Substitution

    solve the following differential equation with the suggested change of variables.
  46. 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...
  47. 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...
  48. 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...
  49. 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.
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