differential equation Definition and Topics - 143 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. potatocake

    How do I solve this differential equation?

    (x cos(y) + x2 +y ) dx = - (x + y2 - (x2)/2 sin y ) dy I integrated both sides 1/2x2cos(y) + 1/3 x3+xy = -xy - 1/3y3+x2cos(y) Then I get x3 + 6xy + y3 = 0 Am I doing the calculations correctly? Do I need to solve it in another way?
  2. F

    Current through Ballistic 2DEG Channel

    So I am a bit uncertain what approach is best for solving this problem and how exactly I should approach it, but my strategy right now is: 1. Solve the time-independent Schrödinger Equation with the given Hamiltonian and find energy eigenvalues of system: -Here I struggle a bit with actually...
  3. Hamiltonian299792458

    I Solving and manipulating the damped oscillator differential equation

    the differential equation that describes a damped Harmonic oscillator is: $$\ddot x + 2\gamma \dot x + {\omega}^2x = 0$$ where ##\gamma## and ##\omega## are constants. we can solve this homogeneous linear differential equation by guessing ##x(t) = Ae^{\alpha t}## from which we get the condition...
  4. Celso

    Superposition in separation method of variables

    Each different boundary condition means a different charge configuration, how can this problem be solved using superposition?
  5. K

    Help solving this Heat Equation please

    I want to solve the heat equation below: I don't understand where the expression for ##2/R\cdot\int_0^R q\cdot sin(k_nr)\cdot r \, dr## came from. The r dependent function is calculated as ##sin(k_nr)/r## not ##sin(k_nr)\cdot r##. I don't even know if ##sin(k_nr)/r## are orthogonal for...
  6. diazdaiz

    Find the function of velocity of a person that accelerates to a constant v

    for example, I want to know velocity of a person when time is equal to t, that person start running from 0m/s (t=0s) to max velocity of 1m/s (t=1s). I am thinking that this is like rain droplet that affected by gravity and drag force, where force is directly proportional to its velocity, to make...
  7. derya

    A Generic Solution of a Coupled System of 2nd Order PDEs

    Hi! I am looking into a mechanical problem which reduces to the set of PDE's below. I would be very happy if you could help me with it. I have the following set of second order PDE's that I want to solve. I want to solve for the generic solutions of the functions u(x,y) and v(x,y). A, B and C...
  8. S

    Classical mechanics -- Equations for simulating the motion of a body

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    • s1mos_tsr
    • Thread
    • clasccal mechanics differential equation mechanical engenereeing physics
    • Replies: 7
    • Forum: Mechanics
  9. J

    A Nonlinear Wave Equation (Nonlinear Helmholtz)

    I am trying to solve a PDE (which I believe can be approximated as an ODE). I have tried to solve it using 4th Order Runge-Kutta in MATLAB, but have struggled with convergence, even at an extremely high number of steps (N=100,000,000). The PDE is: \frac{\partial^2 E(z)}{\partial z^2} +...
  10. H

    Air flow in a balloon

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  11. K

    TISE solution for a hydrogen atom

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  12. S

    I How does one solve Uxx+Uyy+Uzz=C when C is non-zero?

    How does one solve the partial differential equation Uxx+Uyy+Uzz=C when C is non-zero. Here U is a function of x,y and z where (x,y,z) lies in the ball centered at 0 of radius 1 and U=0 on the boundary. Uxx, Uyy and Uzz denote second partial derivatives with respect to x, y and z. Any hints on...
  13. J

    Modeling the populations of foxes and rabbits given a baseline

    From solving the characteristic equations, I got that ##\lambda = 0.5 \pm 1.5i##. Since using either value yields the same answer, let ##\lambda = 0.5 - 1.5i##. Then from solving the system for the eigenvector, I get that the eigenvector is ##{i}\choose{1.5}##. Hence the complex solution is...
  14. A

    I Constants at the end of the Frobenius method

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  15. TheMercury79

    I Scale factor from Friedmann's equations

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  16. Luke Tan

    I How do I solve this ODE?

    When reading through Shankar's Principles of Quantum Mechanics, I came across this ODE \psi''-y^2\psi=0 solved in the limit where y tends to infinity. I have tried separating variables and attempted to use an integrating factor to solve this in the general case before taking the limit, but...
  17. R

    I Does this ODE have any real solutions?

    The ODE is: \begin{equation} (y'(x)^2 - z'(x)^2) + 2m^2( y(x)^2 - z(x)^2) = 0 \end{equation} Where y(x) and z(x) are real unknown functions of x, m is a constant. I believe there are complex solutions, as well as the trivial case z(x) = y(x) = 0 , but I cannot find any real solutions. Are...
  18. GlassBones

    Change of variables on autonomous systems solutions

    Homework Statement Given that ##x=\phi (t)##, ##y=\psi(t)## is a solution to the autonomous system ##\frac{dx}{dt}=F(x,y)##, ##\frac{dy}{dt}=G(x,y)## for ##\alpha < t < \beta##, show that ##x=\Phi(t)=\phi(t-s)##, ##y=\Psi(t)=\psi(t-s)## is a solution for ##\alpha+s<t<\beta+s## for any real...
  19. E

    I Differential Equation Problem

    So i've been Stuck On this Equation Trying to find the integrating factor (im not sure if it has one) appreciate the help
  20. W

    Finding the parametric equation of a curve

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  21. Phys pilot

    Critical points of a diff eq system

    Homework Statement I have to calculate the critical points of the following system. $$x'=cx+10x^2$$ $$y'=x-2y$$ The Attempt at a Solution So I solve the system $$cx+10x^2=0$$ $$x-2y=0$$ So if $$x=2y$$ I have $$2yc+10*4y^2=2yc+40y^2=y(2c+40y)=0$$ and I get $$y=0$$ and $$y=-\frac{c}{20}$$ f I...
  22. W

    Finding Orthogonal Trajectories (differential equations)

    Homework Statement Find Orthogonal Trajectories of ##\frac{x^2}{a}-\frac{y^2}{a-1}=1## Hint Substitute a new independent variable w ##x^2=w## and an new dependent variable z ##y^2=z## Homework Equations The Attempt at a Solution substituting ##x## and ##y## I get...
  23. S

    I Y'' + y = 0 solution and recursion relation

    I've found the general solution to be y(x) = C1cos(x) + C2sin(x). I've also found a recursion relation for the equation to be: An+2 = -An / (n+2)(n+1) I now need to show that this recursion relation is equivalent to the general solution. How do I go about doing this? Any help would be...
    • Sam D
    • Thread
    • differential equation general solution ordinary differential equation power series recursion
    • Replies: 3
    • Forum: Differential Equations
  24. RpWinter

    Solving the differential equation of planetary motion

    Hey, this is how i tried solving the differential equation The solution however does not match the general solution of the equation. Also differentiating it twice does not give me the previous equation. Please tell me if i did some mistake while solving. I already know how to solve by finding...
    • RpWinter
    • Thread
    • differential equation planetary motion
    • Replies: 3
    • Forum: Mechanics
  25. B

    Variation of Parameters to solve a second order ODE

    Homework Statement The question I am working on is the one in the file attached. Homework Equations y = u1y1 + u2y2 : u1'y1 + u2'y2 = 0 u1'y1' + u2'y2' = g(t) The Attempt at a Solution I think I have got part (i) completed, with y1 = e3it and y2 = e-3it. This gives a general solution to the...
  26. I

    I Drude Model Permittivity Formula - e^iwt or e^(-iwt)?

    I've been having a sign problem while deriving the permittivity formula using Drude model, and I found out that the problem came from the fact that complex field vectors are expressed with e-iwt, not eiwt, thus producing (-iwt) term when differentiated...
  27. C

    Calculations for a capacitor charging from another capacitor

    Hi all, I'd appreciate help in calculating the voltages in the circuit shown. I thought it should be fairly straight forward, but it has me stumped. This is a sample-and-hold circuit for an ADC -- the switch is closed to charge the hold capacitor with the sample voltage and then opened to...
  28. 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...
  29. 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...
  30. 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...
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