What is Differential equation: Definition and 1000 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. David Koufos

    Air resistance differential equation

    Hello all, I want to say thank you in advance for any and all advice on my question. My classical mechanics textbook (Marion Thornton) has been taking me through motion for a particle with retarding forces. The example it keeps giving is: m dv/dt = -kmv which can be solved for: v = v0e-kt...
  2. cg78ithaca

    A Generalization of hypergeometric type differential equation

    I am aware that hypergeometric type differential equations of the type: can be solved e.g. by means of Mellin transforms when σ(s) is at most a 2nd-degree polynomial and τ(s) is at most 1st-degree, and λ is a constant. I'm trying to reproduce the method for the case where λ is not constant...
  3. A

    Equations for three inductors connected together

    I have a system with three inductors connected together at a common point. The unconnected ends of each inductor is connected to an independent voltage source. Basically I want to get three expressions for the dynamics of the currents with V1, V2 and V3 as inputs. i.e. i need to eliminate the...
  4. R

    A Asymptotic solutions to a differential equation

    I am asked to solve the differential equation $$ f''(\eta)+\frac{f'(\eta)}{\eta}+\Big(1-\frac{s^2}{\eta^2}\Big) f(\eta) - f(\eta)^3 = 0, $$ for small ##\eta## and large ##\eta## under the condition ##f(\eta \rightarrow \infty) = 1## and ##f(0)=0##. The numerically solved solution looks like...
  5. J

    Partial Differential Equation with square roots

    <Moderator's note: Moved from a technical forum and thus no template.> Hi everyone, I have encountered a partial differential equation with square roots which I don't have a clue in solving it. After letting z=F(x)+G(y), I can't really figure out the next step. I tried squaring both sides but...
  6. Peter Alexander

    Solving Partial Differential Equation

    1. The problem statement, all variables, and given/known data Task requires you to solve a partial differential equation $$u_{xy}=2yu_x$$ for ##u(x,y)##. A hint is given that a partial differential equation can be solved in terms of ordinary differential equations. According to the solution...
  7. L

    I Second order ordinary differential equation to a system of first order

    I tried to convert the second order ordinary differential equation to a system of first order differential equations and to write it in a matrix form. I took it from the book by LM Hocking on (Optimal control). What did I do wrong in this attachment because mine differs from the book?. I've...
  8. G

    A Converting a Vector Differential Equation in Fluid Mechanics

    Hi guys, I have encountered a problem in fluid mechanics that gives a three-dimensional vector differential equation \begin{equation} a \vec{f} + \nabla{a} + b \nabla{c} = \vec{0} \end{equation} where a, b, and c are unknown scalar functions of three-dimensional space and f is a known vector...
  9. Voq

    I Linear differential equation

    a) y'' + 3y' - y = 3sin3x This is not homogeneous. b) y'' + 3y' - y = 0 This is homogeneous. I see b) is homogeneous because it equals to 0. What are further conclusions for that. How we can predict particular solution in a) to be: y = Asin3x + Bcos3x? And how to predict solutions for other...
  10. R

    X-ray Flux density and a differential equation for photon scattering

    Homework Statement Consider interactions of a X-ray beam at a depth, x, within a material. The flux density is: density flux = $$\frac{I}{A}$$ where I is the intensity of the beam that cross a unit area A at right angles to the beam. Let dx be a small slice at the depth x and let dI(x) be the...
  11. K

    B What's the difference between 1000e^0.05t and 1000*1.05^t?

    We have a population of y = 1000 at year 1980 (call it year 0). Every year the population growth rate is 5% per year. y' shows the growth rate of the y (population). Since the population grows by 5% every year, the growth rate is: y' = 0.05y. This is a simple differential equation. When y(0)...
  12. L

    Laplace transform applied to a differential equation

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  13. B

    Submarine Buoyancy Differential Equation

    Homework Statement A submarine of mass 80 000 kg is floating at rest (neutrally buoyant) at a depth of 200 m in sea water. It starts pumping out sea water from its ballast tanks at a rate of 600 litres per minute, thus affecting both its mass and the buoyancy force. Determine the vertical...
  14. Eswin Paul T

    MATLAB Sovling Bernoulli's differential equation in matlab?

    I have a first order bernoullis differential equation. I need to solve this in matlab. Can anyone help me?
  15. E

    2nd-order Nonhomogeneous Differential Equation

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  16. yecko

    2nd order differential equation with power series

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  17. F

    Substitution in a differential equation, independent variable

    Homework Statement $$y'=-\frac{1}{10}y+(cos t)y^2$$ when doing substitute for ##z=\frac{1}{y}## I understand this is ##z(t)=\frac{1}{y(t)}## I know t is independent variable and y is dependent variable but I want to know what is z role here, is it change the dependent variable? when...
  18. D

    How Can I Solve a Differential Equation in Physics Homework?

    Homework Statement In my physics homework, I ran into a differential equation. I am attempting to solve this differential equation for y(x). Homework Equations y''[x] = -C/(y[x]^3) - y[x] C is a constant The Attempt at a Solution dy^2/dx^2 = -C y[x]^-3 - y[x] (1)/(-Cy[x]^-3 - y[x]) dy^2=...
  19. Michael Nelo

    Troubleshooting Nonlinear Pendulum Movement with Air Friction: Tips and Analysis

    The Problem So, I know the basic, the basic differential equation of movement in terms of the angle it is formed, which has the form: 0 = g⋅sin[α(t)] + α''(t)⋅l. However, I decided to consider the friction force of the air, which is always contrary to the movement, and proportional to the...
  20. T

    MHB Non-homogeneous differential equation, undetermined coefficients

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  21. R

    When is the tank empty? [differential equation]

    Homework Statement problem 23: A large tank is filled to capacity with 500 gallons of pure water. Brine containing 2 lbs of salt per gallon is pumped into the tank at a rate of 5 gal/min. The well mixed solution is pumped out at the same rate. Find the number A(t) of lbs of salt in the tank at...
  22. R

    Find the general solution of the given differential equation....

    Homework Statement Find the general solution of the given differential equation. Give the largest interval I over which the interval is defined.Determine wether there are any transient terms in the general solution 5. \frac {dy}{dx} + 3x^2y = x^2 Homework EquationsThe Attempt at a Solution...
  23. JTC

    A Causality in differential equations

    Hello, I am studying control theory. And I have encountered something I have never considered or thought about. Consider a system with y as the output differential equation and u as the input. any(n) + ... + a1y(1) + a0y = bmu(m) + ... + b1u(1) + b0u Here, the subscripts indicate...
  24. O

    Motion of a non-linear pendulum with air resistance

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  25. V

    What went wrong with my simple differential equation?

    Homework Statement [/B] dy/dt = c - ky Homework Equations integral 1/y dy = ln(y) The Attempt at a Solution let y = c/k + z dy/dt = dz/dt = -kz dz/z = -kdt ln(z) = - kt z = e^(-kt) but z = y - c/k y = e^(-kt) + c/k + cons. answer should have been negative sign on the e term. I...
  26. Dor

    A Numerical solution for a two dimensional 3rd order nonlinear diff. eq.

    I'm a bit lost in all the numerous methods for solving differential equations and I would be very grateful if someone could point me to some direction. I want to solve the following boundary conditioned differential equation: $$a_1+a_2\nabla f(x,y)+a_3\nabla f(x,y)\cdot \nabla^2...
  27. Pushoam

    Solving 2nd order differential equation

    Homework Statement Homework EquationsThe Attempt at a Solution For the homogeneous equation, I have got the the root of the characteristic equation as ## e^{ix}, e^{-ix} ## . So, the corresponding solution is ## B \sin{ x} + A \cos{ x} ## . Then, I took the particular solution as...
  28. T

    I Solving for constants in a differential equation

    I feel so sorry when I found myself trapped in a basic problem like this one, but let's go ahead... Suppose we have the following equation, knowing that ##B## is a constant, $$\frac{dU( \theta)}{d \theta} + 2Br = 0$$ where we want to solve for ##B##. If we differentiate the above equation with...
  29. Alexander350

    B Solving a differential equation with a unit vector in it

    I need to solve: \dot{\mathbf{r}}=-kv\hat{r} - \dot{\mathbf{r}_s} However, I do not know how to deal with the fact that there is a unit vector. How can this be done? \dot{\mathbf{r}_s} is a constant vector.
  30. E

    MHB Differential equation w/ cos and sin

    It would be wonderful if someone could please help with the following question as I don't even know where to begin y=y(x), where x^2 cos y + sin(3x-4y) =3Thank you :)
  31. M

    First-order differential equation

    Homework Statement y'+tanxy=sinx Homework Equations integrating factor I(x)= exp{lnIsecxI}[/B]The Attempt at a Solution I have secxy= integral of sinx I(x) I am not sure how to integrate that because secx is in absolute value form.[/B]
  32. B

    Solving a differential equation with substitution

    This is a small part of a question from the book, so I think the format does not really apply here. When doing questions for solving differential equation with substitution, I encountered a substitution ## y(x)=\frac{1}{v(x)} ##. And I am not sure about the calculus in finding ## \frac{dy}{dx}...
  33. ecoo

    Differential Equation, Rewriting solution

    Homework Statement (also express C and alpha as functions of A and B) I need help with the second part (rewriting the solution). Homework Equations ejθ = cos(θ) + jsin(θ) The Attempt at a Solution Unfortunately, I can't think of how to even begin solving. I have the notion that I have...
  34. J

    Finding the singular points for this differential equation

    Homework Statement If d^2/dx^2 + ln(x)y = 0[/B]Homework Equations included in attempt The Attempt at a Solution I was confused as to whether I include the power series for ln(x) in the solution. It makes comparing coefficients very nasty though. Whenever I expand for m=0 for the a0 I end...
  35. P

    MHB Eugene's question via Facebook about a Differential Equation

    This equation is separable... $\displaystyle \begin{align*} \frac{\mathrm{d}y}{\mathrm{d}x}&= 3\,\sqrt{4 - y^2} \\ \frac{1}{\sqrt{4 - y^2}}\,\frac{\mathrm{d}y}{\mathrm{d}x} &= 3 \\ \int{ \frac{1}{\sqrt{4 - y^2}}\,\frac{\mathrm{d}y}{\mathrm{d}x} \,\mathrm{d}x} &= \int{ 3\,\mathrm{d}x} \\ \int{...
  36. C

    Differential Equation for Projectile Motion with Air Drag

    Hi! I was wondering how I could come up with a differential equation for projectile motion on a 2D plane when air resistance is not negligible. I'm trying to guess the position of a projected ball at a certain time period by approximating the coordinates using the Euler's method. Here, I would...
  37. F

    Partial differential equation boundary

    Homework Statement I have to calculate the stationary field inside a room. Homework EquationsThe Attempt at a Solution I used the diffusion equation to calculate the temperature, which is T(x,y)=(Eeknx+Fe-knx)cos(kny), k=(n*pi/a), a is the length of the room. Now i have to satisfy boundary...
  38. ecoo

    Momentum Differential Equation

    Homework Statement A rocket sled moves along a horizontal plane, and is retarded by a friction force friction = μW, where μ is constant and W is the weight of the sled. The sled’s initial mass is M, and its rocket engine expels mass at constant rate dM/dt ≡ γ; the expelled mass has constant...
  39. F

    Projectile Motion: Angle & Range with Air Resistance and Wind

    Homework Statement A projectile is launched at an angle to the horizontal and rises upwards to a peak while moving horizontally. What is the angle of maximum range and how it is dependent on initial velocity if we include air resistance and if the wind is blowing in the horizontal direction of...
  40. S

    Wave properties from the differential equation of a wave

    How can we work out all the properties of wave from differential equation? And what really does differential equation of wave implies?
  41. S

    Engineering DC motor differential equation

    This question involves finding the transfer function for the system, but I first need to get the differential equations correct. Have I set up the gearbox correctly?
  42. A

    Solving partial differential equation with Laplace

    Homework Statement am trying to solve this PDE (as in the attached picture) https://i.imgur.com/JDSY4HA.jpg also my attempt is included, but i stopped in step, can you help me with it? appreciated, Homework EquationsThe Attempt at a Solution my attempt is the same as in the attached picture...
  43. Telemachus

    I Separation of variables for nonhomogeneous differential equation

    Hi. I was wondering if it is possible to apply separation of variables for a function of space and time obeying a non homogeneous differential equation. In particular, the heat equation: ##\displaystyle \frac{\partial \Phi(\mathbf{r},t)}{\partial t}-\nabla \cdot \left [ \kappa(\mathbf{r})...
  44. Gh. Soleimani

    A The differential equation of Damped Harmonic Oscillator

    If you consider b^2/m > 4*k, you can get the solution by using classic method (b = damping constant, m = mass and k = spring constant) otherwise you have to use complex numbers. How have the references books proved the solution for this differential equation?
  45. 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...
  46. grquanti

    I Substitution in partial differential equation

    Hello everybody. Consider $$\frac{\partial}{\partial t}f(x) + ax\frac{\partial }{\partial x}f(x) = b x^2\frac{\partial^2}{\partial x^2}f(x)$$ This is the equation (19) of...
  47. S

    Differential Equation Resonance

    Homework Statement I was reading a PDE book with a problem of resonance $$ y_{tt} (x,t) = y_{xx} (x,t) + A \sin( \omega t) $$ After some work it arrived to a problem of variation of parameters for each odd eigenvalue. To solve it, it uses $$ y''(t)+a^{2} y(t) = b \sin ( \omega t) \qquad y(0)=0...
  48. Adgorn

    I Question regarding integration of an equation

    I am beginning to learn about differential equations and I saw in an explanatory video a solution to this separable differential equation: ##\frac {dy} {dx} = \frac {-x} {y{e^{x^2}}}## from there through simple steps the equation changed to ##y⋅dy=-xe^{-x^2}⋅dx ##. Then the video did an...
  49. Salvador_

    Classical mechanics differential equation F(x) = -kx

    Homework Statement A particle of mass m is subject to a force F (x) = -kx. The initial position is zero, and the initial speed is v0. Find x(t). Homework Equations F = m*v*dv/dx = -kx v = dx/dt The Attempt at a Solution I'm new to differential equations, so please excuse me if I make any...
  50. Drakkith

    I When is a Constant a Solution to a Differential Equation

    I've run across several instances while doing homework where a question will have two solutions. One will be an equation, and the 2nd will be a constant (usually zero). I can't figure out why this constant is a solution though. For example, take the following differential equation...
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