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. W

    Dirichlet Problem-Partial Differential Equations

    Homework Statement Let K = \{ (x,y) | -1<x<1 , -1<y<1 \} . Find the unique soloution of dirichlet problem: \Delta u(x,y) =0 , (x,y) \in K , u(x,y) = |x+y| , (x,y) \in \partial K . Homework Equations The Attempt at a Solution We need to guess a soloution and not use...
  2. J

    Higher-order differential equations

    Homework Statement y'' + 2y' +y = 5e^6x use undetermined coefficients to solve this equation. Homework Equations The Attempt at a Solution what I would normally do is m^2 + 2m + 1 = 0 but then I get stuck I can't find the general equation as this won't factor. I tried...
  3. L

    Differential Equations problem

    Homework Statement dx/dy = cos(y) - xtan(y) I need to find the general solution of the problem Homework Equations y' + Py = Q Where P and Q are functions of x dy/y = - Pdx ln(y) = -integral(Pdx)+c y = e^(-int(Pdx)+c) The Attempt at a Solution Now I have no idea...
  4. S

    How to Solve a System of Coupled Differential Equations?

    Homework Statement [b]2. Homework Equations [/bHomework Statement I have to find a(t) and b(t), using the following set of equations: \frac{\hbar\omega_{1}}{2}a(t)\cos{\alpha}+\frac{\hbar\omega_{1}}{2}b(t) exp(-i\omega t) \sin{\alpha}=i\hbar\frac{da}{dt} \frac{\hbar\omega_{1}}{2}a(t)...
  5. J

    Solution for Differential Equations with x = e^t

    Homework Statement ax^{2}\frac{d^{2}y}{dx^{2}}+bx\frac{dy}{dx}+cy=0 Let x= e^{t} Find \frac{dy}{dx} and \frac{d^{2}y}{dx^{2}} in terms of \frac{dy}{dt} and \frac{d^{2}y}{dt^{2}} The Attempt at a Solution if x= e^{t} then \frac{dx}{dt} = e^{t}= x \frac{dy}{dx}= \frac{dy}{dt}\frac{dt}{dx}=...
  6. S

    Solving differential equations in four variables

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  7. H

    System of differential equations eigenvalues

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  8. C

    Related Rates - Differential Equations

    Homework Statement A ladder, of length 13m, is leaning against a wall. The base of the ladder begins to slide, frictionless, away from the wall; by the time the base is horizontally 12m away from the wall it is moving at a rate of 2ms-1 a)How fast is the top of the ladder sliding down the...
  9. F

    Solving Differential Equations: the Euler D.E. and Linear D.E.s

    Hello guys, I am new here to the forum and I was wondering if you could help me with some trouble I am having with differential equations. For your information: I am a second year applied physics student from the Netherlands and I loveee everything about physics and mathematics! The problem...
  10. B

    In dealing with differential equations

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  11. E

    Differential Equations Inverse Laplace(Partial Fractions)

    1. L-1{(3s+2)/ (s2+2s +10)} After completing the square I get to 3s+2 /(s+1)2 + 32 which suggests two solutions or one. They decompose the fraction into [(A)s+1 /(s+1)2 + 32 ]+ [(B) 3/(s+1)2 + 32] I am unsure of how this decomposition works I thought that we would take A(3s) as the numerator...
  12. O

    Minimal Surfaces, Differential Geometry, and Partial Differential Equations

    Last night in a lecture my professor explained that some partial differential equations are used to observe events on minimal surface (e.g. membranes). A former advisor, someone that studied differential geometry, gave a brief summary of minimal surfaces but in a diffy G perspective. 1.)...
  13. A

    Differential equations - undetermined coefficients

    Homework Statement Find a particular solution of y'' + 2y' + 5y = e^x * sin(x)Homework Equations The Attempt at a Solution Characteristic equation: r^2 + 2r + 5 = 0. Through quadratic formula, r1 = -1+2i and r2 = -1-2i, so my homogeneous equation is y_h = c1 * e^(-x) cos(2x) + c2 * e^(-x) *...
  14. A

    Differential equations - variation of parameters

    Homework Statement Find a particular solution using variation of parameters. y'' + 3y' + 2y = 4e^x Homework Equations yp = -y1 * INT (y2f(x)/W[y1,y2]) dx + y2 * INT (y1f(x)/W[y1,y2]) dx The Attempt at a Solution So, first I find the homogeneous solution, correct? r2 + 3r + 2 = 0, so...
  15. B

    Finding the Second Solution for a System with a Repeated Eigenvalue

    Homework Statement I'm told to solve the following: X' = (1 1; -1 3)X Where (1 1) is the first row and (-1 3) is the second row. The Attempt at a Solution Okay so I calculated teh eigenvalues and I got repeated eigenvalue of 2. I calculated the eigenvector and got (1,1). So a solution...
  16. D

    Differential equations stuck trying to integrate 10^-u du

    Homework Statement trying to find the integral curve for ths equation get stuck at trying to integrate 10^-u dy/dx=10^x+yHomework Equations The Attempt at a Solution let u=x+y so dy/dx=10^u du/dx=1+dy/dx =(10^u)-1=du/dx du/dx=(10^u)-1 du/10^u=-dxthanks in advance
  17. N

    Solve Differential Equations problem

    Solve: dx/dt=3-x initial condition x(1)=4.
  18. C

    Quick Question on Variation of Parameters Differential Equations

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  19. M

    Solving Differential Equations

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  20. V

    Critical Points and Differential Equations Helppp

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  21. D

    MATLAB Help with Matlab solving second order differential equations

    I am a Matlab rookie. I need to solve numerically the following second order differential equations d^2x/dt^2 + w0_(el) * x = e/m_e * E - K3/m_e * x *y; d^2y/dt^2 + w0_(v) * y = - K_3/2M * x^2; I have started to deal with only the harmonic part of the problem. So I tried to solve...
  22. B

    MATLAB, Differential Equations help

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  23. S

    Matlab, differential equations

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  24. M

    LaPlace Transformations to Solve Ordinary Differential Equations

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  25. R

    Linear Differential Equations, no x in p(x) or q(x)?

    An integrating factor e^(integral p(x) ) can be used to solve Linear D.E. in form y' + P(x)y=Q(x), but what do i do when there is no x in P(x) or Q(x)? for instance in these problems: #1)y' - y=4(e^x) #2)y'+2xy=4 Is it wrong to use P(x)= -1 for problem 1, and Q(x)=4 for problem 2? This...
  26. H

    Population Equilibrium Solutions Differential Equations

    Homework Statement Consider the following model for the populations of rabbits and wolves (where R is the population of rabbits and W is the population of wolves). dR/dt = .05R(1-.00025R)-.00109375RW dW/dt = -.04W+8e-05RW Find all the equilibrium solutions: In the absence of...
  27. M

    Riccati differential equations

    riccati differential equations -------------------------------------------------------------------------------- how to find general solution of this question du/dt=u^2+t^2 please say me i work hard but i do nat know this form of riccati equation. i know when special solution is...
  28. M

    Differential Equations Assignment (small step needs clarification)

    When given y'' + ay' + by + c = 0 The characteristic equation is: λ^2 + aλ + b = 0 Now I'm making a substitution of e^(λx) to get [(λ^2 + aλ + b)][e^(λx)] = 0 double root : [(λ - alpha)^2] [e^(λx)] = 0 My question now is that if I were to do the same thing with the...
  29. M

    Second Order Linear Nonhomogeneous Differential Equations

    Homework Statement y''+7y'=392sin(7t)+686cos(7t) with y(0)=4 and y'(0)=9 Homework Equations No real relevant equations The Attempt at a Solution I assumed since the g(t) has function of both sine and cosine the solution would be both the real and non real parts of the solution to...
  30. A

    Differential equations - nonhomogeneous

    Homework Statement Let Ly = y'' + py' + qy. Suppose y1 and y2 are functions such that Ly1 = f(x) and Ly2 = g(x). Show that the sum y = y1 + y2 satisfies the nonhomogeneous equation Ly = f(x) + g(x). Homework Equations Superposition Principle: L[c1y1 + c2y2] = c1L[y1] + c2L[y2] The...
  31. W

    Help with non-homogeneous ordinary differential equations

    Homework Statement I have a Driven Harmonic Oscillator Without a Damping Force Homework Equations m\ddot{}x+kx=F0 cos(\omegat)The Attempt at a Solution Not looking for a solution just looking for an example problem, as none are provided in my text. If anyone can provide one for me I'd be very...
  32. C

    Second order linear differential equations nonhomogeneous equations

    I could not get LaTex to format properly, so I typed out the question and my work using Microsoft Word's equation editor. Please see the attached document, apologies for any inconvenience! These problems involve the techniques for the method of undetermined coefficients and variation of parameters.
  33. T

    Differential equations question - possible integrating factor?

    Homework Statement Solve the differential equation with the initial condition y(0) = 1 Homework Equations 3e5xdy/dx = -25x/y2 The Attempt at a Solution First I tried putting everything with an x or dx on one side and a y or dy on the other side, and solved for C. I got 0 as the...
  34. P

    How to Solve Differential Equations for Homework?

    Homework Statement see attached Homework Equations The Attempt at a Solution Question 7a) I can do but 7b I am not sure how to start. I think x with a dot means differential with respect to time? need some help starting. Thanks
  35. H

    Differential Equations initial value problem

    Homework Statement Let f(t) be the solution to the initial value problem 2t(dy/dt)+y=t^4 with f(0) = 0 find f(t). Homework Equations The Attempt at a Solution I tried to do this by separating variables but that hasn't gotten me very far. I don't know if I can do it by doing...
  36. Saladsamurai

    System of Differential Equations

    Homework Statement \frac{dx}{dt} +\frac{dy}{dt} + y - x = e^{2t} \frac{d^2x}{dt^2} +\frac{dy}{dt} = 3e^{2t} I am a little stumped on this one. I tried writing it using D as the differential operator d/dt: D(x+y) +x - y = e2t D2x+Dy = 3e2t but that doesn't do anything but...
  37. T

    Solving the simples Partial Differential Equations

    Hello, I joined a class on Partial Differential Equations 3 or 4 lectures late, so I have missed the classes outlining how to solve the simplest forms of them. I am trying to understand the textbook, but so far it is not going so well for me, so I just need to ask some questions about the...
  38. S

    System of Differential Equations

    Homework Statement So my friend asked me for help because he assumed having a math degree meant I knew math :rolleyes: [PLAIN]http://courses.webwork.maa.org:8080/wwtmp/equations/0e/c5a05957810916cfdff379ee0642fc1.png (dx/dt=−4y and dy/dt = −4x) Convert this system to a second...
  39. M

    MATLAB Solving Matrix Differential Equations with ODE45

    If we have differential variable of matrix kind ( with dimention 3*3 ) and non matrix ( with dimention 1) simaltanously and as a couple ,how can we use ode45 ? If we have only matrix kind , we can use m file of ode at first with the command of reshape (3,3) and then change it to column one.What...
  40. O

    Partial differential equations

    can anyone help me to solve this equation ?. is this equation linear or not? knowing, this equation represents the equation of motion for frame members for materials which have different moduli in tension and compression (civil engineering). "EI" ("x,t" ) ("∂" ^"4" "u" ("x,t" ))/〖"∂x" 〗^"4"...
  41. H

    Differential Equations mixing sugar and water

    Homework Statement A tank contains 2860 L of pure water. A solution that contains 0.04 kg of sugar per liter enters a tank at the rate 5 L/min The solution is mixed and drains from the tank at the same rate. Find the amount of sugar in the tank after t minutes. Homework Equations...
  42. Saladsamurai

    Why do we do this in Differential Equations?

    I have always gone along with this, but now I am not so sure: Take the DE: y' +4y = 0 If you carry out the integration, we get y = e^{-4x} +C\qquad(1) but we always pull some stuff like "well since we could write C as ec1, since it's arbitrary", we can write the solution like...
  43. D

    Solving driven differential equations

    Homework Statement The suspension system of a car is designed so that it is a damped system described by z'' + 2z' + z = f(t) where z is the vertical displacement of the car from its rest position. The car is driven over a (smooth!) road which has "catseye" embedded in the road surface...
  44. A

    Ordinary differential equations with boat

    Question: A boat of mass m is traveling with the velocity v0. At t=0 the power is shut off. Assuming water resistance proportional to v^n, where n is a constant and v is the instantaneous velocity, find v as a function of the distance travelled. (Note that you need to consider the two cases)...
  45. K

    Online Calculus 2 / Differential Equations for Credit?

    Hello all! As it turns out, I have a condition that requires me to not be at a physical university this semester, but I would like to try and catch up a bit and also need to make my time off more productive. I would like to take a couple of math courses, particularly Calculus 2 and...
  46. T

    Ordinary differential equations

    Find equilibrium solutions for the following ODE initial-value problem and linearize the problem about those solutions z" = z − z^3, z(0) = z0 z'(0) = v0 i found the equilibrium solutions to be 0,1,-1. what are the steps to linearize the problem around these?
  47. S

    RC Circuits (Differential Equations)

    Homework Statement Find the voltage across a capacitor in an RC Circuit, using [V]_{}[/c] (0) = 1, [V]_{}[/i] n(t)=t. Homework Equations dV/dt = (1/RC)(V)=(1/RC)([V]_{}[/in]) The Attempt at a Solution New to this site: I honestly don't know where to start. Done well in Calculus...
  48. N

    System of differential equations, Maple tells me I'm wrong

    Hi there, I have a system of differential equations which I set up in a matrix like this: \left[ \begin {array}{ccc} 3/2&-1&-1/2\\ \noalign{\medskip}-1/2&2&1/2 \\ \noalign{\medskip}1/2&1&5/2\end {array} \right] = \left[ \begin {array}{c} {\frac {d}{dt}}x \left( t \right) \\...
  49. I

    Differential Equations - not Linear, Separable, or Exact

    Problem and Equation: Solve dy/dx=-y/(x^2+y) Put into standard form, this is ydx+(x^2+y)dy=0 The only ways of solving differential equations that I currently know are when they are either linear (which this is not), separable (this is also not), or exact (ditto), and I vaguely know about...
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