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. C

    Solution to the differential equation describing a mass-spring-damper

    What is the general solution to the differential equation describing a mass-spring-damper? t=time x= extension of spring M=Mass K=Spring Constant C=Damping Constant g= acceleration due to gravity Spring has 0 length under 0 tension Spring has 0 extension at t = 0 If the Force...
  2. D

    Differential equation, limit cycles, stability, phase plane. By 11/12

    Homework Statement The trajectory of an arrow in space obeys the following system of equations: \dot{x} = y+(x^2+y^2-3)^2 (x^3-x+xy^2) \dot{y} = y+(x^2+y^2-3)^2 (y^3-y+x^2y) 1. Questions a) Derive an ODE for the radial coordiante r(t) = \sqrt[]{x^2(t)+y^2(t)} b) Show that the...
  3. A

    What is the difference between a differential equation and a derivative?

    For example, if y=x^2, then the derivative of y is 2x. We write the derivative as either f'(x)=2x or dy/dx=2x. Well, the differential equation is also written as dy/dx=2x. So is there a difference between a differential equation and a derivative? !~Alshia~!
  4. J

    Finding the differential equation for a oscillating system

    Homework Statement The Attempt at a Solution I don't think this is the correct answer because for some reason I have a constant mg term. Usually I get mgsinθ and from small angle approximations it becomes mgθ, but this time I am getting mgcosθ and from small angle approximations it...
  5. C

    RLC circuit, finding relations and forming a differential equation

    Hello. I'm having a hard time solving this homework question, even though I know I shouldn't... but well. Here's the problem : -The K switch is closed for a long time, so as to be in a steady state. What's the electric charge of the capacitor ? Infer vs tension from this. Which...
  6. T

    Differential Equation system and matrix

    I have a quick question. I have to write the differential equations in matrix form Eq1:= x1'(t) = -a1*x1(t) + vf Eq2:= x2'(t) = a1*x1(t) - a2*x2(t) is this correct :
  7. C

    Differential equation for freely swinging body

    Homework Statement A disk or radius b rotates about a rod in its center with a constant angular velocity, Ω. At the disk's edge A, a pin is attached allowing for a body to be attached an freely swing during the rotation. Determine the differential equation for the angle β between the attached...
  8. R

    Problem with solving differential equation to solve for time

    Homework Statement Oil is released from a submerged source at a constant flow of rate of Q=(0.1 m3)/s. Density of oil (p=870 kg/m3) is less than that of water and since oil is only sparingly soluble in water a slick will form on the water surface. Once its formed it will have a tendency to...
  9. C

    Solving a linear 2. order non-homogeneous differential equation

    This is a case where an object is coupled to a spring, laying on a table. The object is moving, friction less, horizontally on the table. We assume the object is moving in an outer forice field which acts in the same direction as the object's motion. The motion is modeled by y''(t) + y(t) =...
  10. P

    3rd order non linear differential equation

    Q. Find SP of:\dddot{x} + \ddot{x} + \dot{x} = x^3 -2x^2 - 31x -28 x(t)=x And determine of the solutions as stable or unstable. OK, not seen one like this before. I've done it with 2nd order derivatives and wondered if it was the same. by setting the derivatives to zero and solving the RHS...
  11. S

    Simple Pendulum nonlinear second order differential equation

    Homework Statement given: dt=-\frac{1}{2}\sqrt{\frac{l}{g}}\frac{d\theta}{\sqrt{sin^2(\alpha/2)-sin^2(\theta/2)}} make the change of variables sin(\theta/2)=sin(\alpha/2)sin(\phi) to show that: dt=-\sqrt{\frac{l}{g}}\frac{d\phi}{\sqrt{1-k^2sin^2(\phi)}} where k=sin(\alpha/2) Homework...
  12. C

    4th order differential equation

    I'm trying to find the gen. solution to the equation y''''-8y'=0 I found the characteristic polynomial by plugging in ert as a solution to y. I got, r^4-8r=0 I simplified to get r*(r^3-8) Thus one root is 0, for the other 3 i must find the cubed root of 8. I know the answer is...
  13. A

    Differential equation, is book making typo?

    Any idea on how they got from first line to second? They just ate the (-1/9)e^(3x)...(picture is attached)
  14. N

    Using Laplace Transform to solve a differential equation

    Homework Statement y" + y = 4δ(t-2π); y(0)=1, y'(0)=0 Homework Equations L[f(t-a) U(t-a)] = e^{-as} L[f(t)] L[δ(t-c)] = e^{-cs} The Attempt at a Solution My answer is: cos(t) + 4U(t-2π)sin(t-2π). When I used Wolframalpha it gave me 4sin(t)U(t-2π) + cos(t)
  15. B

    Solve bernoulli differential equation with extra constant

    Homework Statement Find a solution for: u'(t)=c*u(t)^2-c*(a+b)*u(t)+c*a*b The Attempt at a Solution I've found the solution for the homogeneous equation: u_0(t)=(\frac{1}{a+b}+d*e^{c(a+b)t)})^{-1} Where c is a random constant. Now I've tried the solution u(t)=x(t)*u_0(t), when I fill...
  16. K

    Series solutions for differential equation

    Homework Statement Use the power series to solve the following differential equations, state the first four terms of the two independent solutions. 3xy'' + y' - y = 0 Homework Equations The power series. The Attempt at a Solution How do I get two independent solutions out of this? All...
  17. T

    Differential equation resembling to cycloid

    What is the function corresponding to this ODE: http://home.arcor.de/luag/math/dgl.jpg In complex notation it obviously shows up like this: a * z''(t) + b * |z'(t)| * z'(t) + c = 0; The numerical solution shows a graph resembling to a cycloid. Thanks for any help! Tom
  18. D

    Differential Equation with growth

    So, I have a differential equation with growth problem. It is a dm/dt function and I need to get it to dD/dt, change in diameter with time. here is the original functon, dm/dt= (pi/4)(D)^2* (V(D))*(LWC)*E E=1 LWC = 2 V(D) = 343D^0.6 m/s it starts from a diameter of 1mm and grows to...
  19. M

    Nonlinear second order differential equation

    I am having a problem finding the solution for this eq: y''(x)+(2/x)y'(x)+(w^2)y(x)=0 I couldn't find examples in the textbook that goes on a similar line, and have been searching the internet as well, but no use. I am thinking of using substitution v=y' but not sure how to do that in the...
  20. T

    Find the Differential Equation Relating Input Voltage and Output Voltage

    Hey all, I'm stuck on a dynamic systems question, it's attached as a jpeg I started off by writing nodal equations for each node: Node 1: 1/R1(ei-eA)=C1D(eA-eo)+1/R2(eA-eB) Node 2:1/R2(eA-eB)=C2D(eB) I know that I have to isolate for ei and eo but I'm really confused with...
  21. N

    Covert differential equation into a system of 1st order ODE?

    How to covert this differential equation into a system of one order ODE? (require covert the equation into a system of 1st-order equations and solve by using ode23 in matlab) x^2*y''-2*x*y'+2*y = 0; y(1) = 4; y'(1)=0; solve for y(x) I tried to convert it get X' = AX in which X...
  22. N

    How to covert this differential equation into a system of one order ODE?

    How to covert this differential equation into a system of one order ODE? x^2*y''-2*x*y'+2*y = 0; y(1) = 4; y'(1)=0; solve for y(x) I tried to convert it get X' = AX in which X = [y, z]' A = [0, 1; 2/x^2, 2/x]; But x exists in A, which cannot solve by dsolve in Matlab.
  23. J

    Differential Equation with Summation

    Homework Statement y''+0.1y'+y=1+2\sum_{k=1}^{n}(-1)^{k}u_{k\pi}(t) and quiescent initial conditions. Homework Equations None. The Attempt at a Solution (s^{2}+0.1s+1)Y(s)=\mathcal{L}\{1\}+2\sum_{k=1}^{n}(-1)^{k}\mathcal{L}\big\{ u_{k\pi}(t)\big\} I'm not sure if this step was...
  24. T

    Simple Differential equation with reduction of order

    use method of reduction of order to find second solution: t2y''-4ty+6y = 0 , y1(t)= t2 Attempt: So I've done all the steps, up to the substitution, but I'm having problems with what appears to be a simple linear equation but I can't solve it: Any ways, with w = v' I arrive at...
  25. C

    Solve differential equation with boundary conditions using substitution

    μ^{2}\frac{d^{2}u}{dx^{2}}+ae^{u}=0 Boundary conditions: u(-L)=u(L)=u_{0} Solve by multiplying by \frac{du}{dx} and integrating in x I know you have to use substitution, but I keep going in circles.
  26. G

    Solving Non-Linear Differential Equation with Fourier Transforms

    Hiya. I have to solve this bad boy under the assumptions that f, f' and f'' tend to 0 as |x| tends to infinity: 1/2(f')^2 = f^3 + (c/2)f^2 + af + b where a,b,c are constants. My thoughts are use Fourier Transforms to use the assumptions given, but not sure how to do them on these terms...
  27. D

    Seemingly Non-Separable Differential Equation

    Homework Statement I am trying to find the parametric equation that describes the following second order differential equation: Homework Equations m\frac{d^2y}{dt^2}=-mg - k\frac{dy}{dt} Where m, g, and k are all constants. The Attempt at a Solution I substituted u=\frac{dy}{dt} to reduce...
  28. F

    L'Hospital's Rule for Limit of x^a ln(x) as x Approaches 0 (a>0)

    Homework Statement Use L'Hospitals rule to show that lim x->0 x^a ln(x) = 0 I don't know how to solve this. I guess the first thing to do is to transform it in some way so that one can use L'Hospitals rule, but I don't know how. Thank you! EDIT: a>0 It's not a differential equation as...
  29. S

    Partial differential equation with conditions

    I'm not sure how to solve this: du/dt = 3 \frac{d^{2}u}{dx^{2}} These are the conditions: u(0,t)= -1 u(pi,t)= 1 u(x,0) = -cos 7x Suggestion: I should use steady state solution to get a homogeneous initial condition. Starting with separtion of variables u(x,t) = G(x)H(t) And...
  30. W

    Integrating Factor for First Order Linear Differential Equation

    Homework Statement Find an integrating factor for the first order linear differential equation \frac{dy}{dx} - \frac{y}{x} = xe^{2x} and hence find its general solution Homework Equations The Attempt at a Solution I found the integrating factor which is e^{-lnx} = x^{-1} and...
  31. J

    Is this a differential equation?

    Mod note: HTML size = ... tags are not needed. You can make things look just fine using [ tex ] tags instead of [ itex ] tags. So, I was doing some stuff, messing around when I thought of something. What if I took a random physics formula and integrated it into the original function? Then I was...
  32. S

    MHB Solving Second order non - homogeneous Differential Equation

    How to solve \( (x+1) y'' - (2x+5) y' + 2y = (x+1) e^x\) can we assume \(y_1 = (Ax+B) e^x \), then \(y_2= vy_1​\) Is this right? then solve for A and B Finally \( y = c_1 y_1 + c_2 y_2\)
  33. iVenky

    Differential equation of a circle

    Consider a circle of radius 'a' and centre (h,b) then the equation of the circle is given by (x-h)2 + (y-b)2 = a2 I expressed this in terms of differential equations which is - a= {[1+(dy/dx)2]3/2}/{d2y/dx2} According to my book - this equation indicates that 'a' is a...
  34. A

    What kind of differential equation is the Schrodinger equation?

    Does it have an easy classification (elliptic, hyperbolic, parabolic, for example)? Or does the fact that it has an "i" in it make this impossible?
  35. C

    A Second Order Differential Equation

    Homework Statement Hi, this problem is from first chapter of Mathematical Methods of Physics by Mathews and Walker. (Problem 1-36, second edition) Consider the differential equation y'' - xy + y^3 = 0 for large positive x. a-) Find an oscillating solution with two arbitrary constants. b-)...
  36. U

    Finding Differential Equation for Circuit: Help Needed!

    Hi I need to try and find the differential equation representing the attached circuit. My work is also being shown on the attachment. Can anyone confirm whether this is correct? If it is wrong could you please provide input as to why? Thanks. Sorry for the quality in advance.
  37. A

    Need explanation of this differential equation

    I need explanations at the last part of this math solution. Question: Solve the differential equation: y' = (1 + 2/x)y Answer: ln|y| = x+ln(x^2)+c |y| = e^c.x^2.e^x y = Cx^2.e^x (C = +/-e^c is any constant that is not equals to 0) What I don't understand is this part where ...
  38. T

    Finding a fundamental set of solutions for a 2nd order differential equation

    Homework Statement 64y''+144y'=0 y1(0)=1 y'1(0)=0 and y2(0)=0 and y'2(0)=1 Homework Equations y1=c1*e^(r1*t) + c2*e^(r2*t)The Attempt at a Solution I start by finding the characteristic equation: 64r^2+144r=0 r1=-9/4 and r2=0 y1=c1e(r1*t) + c2e(r2*t) so I get y1=c1e^(-9/4 *t) + c2e^(0*t)...
  39. C

    MATLAB Differential equation for matlab

    THE PROBLEM : y(t) = e^(-t)*sin(t^2); with t0 = 0 and T = 3.14159. Find y_0, and use it to deduce the corresponding expression for f(t, y) (Your f should have both a t and a y in it. Simplify it to find the y!). This is for a MATLAB project. I've solved this differential equation (which we...
  40. S

    MHB Solving Second order non - homogeneous Differential Equation

    To Solve y’’ – 2 y’ – 3y = 64 e-x x ---------------(1) Using the method of undetermined coefficients : The roots of the homogeneous equation are 3 and -1, so the complimentary solution is y = c1 e3x + c2 e-x Then the guess for the particular solution of (1) is e-x x (Ax + B)...
  41. P

    What Methods Solve the Differential Equation f '(x) = 3 * f(x)?

    Can anyone give some clues on how to solve this differential equation: f '(x) = 3 * f(x) Thanks!
  42. M

    Differential Equation, nonlinear, nonexact

    Homework Statement \frac{dy}{dx}=\frac{2y - x + 7}{4x - 3y -18} Homework Equations The Attempt at a Solution I tried using v = y/x and got nothing. Same goes for trying to find an integrating factor to make the equation exact. I am given a hint, Find h and k so that the...
  43. A

    MHB Solve Second Order Differential Eq. With Variable Coefficient

    $(1-x^2)y'' - xy' + 4y =2 x \sqrt{1-x^2} $ Hint use the substitution $x =\sin t$ I used it and end with $\cos t y'' + \sin t y' - \frac{\sin t}{\cos t} y' + 4y = 2\sin t |\cos t| $ how to solve this i just want the name of the method
  44. M

    Solving the Differential Equation

    I'm given the differential dy/dx = x*y*sinx / (y+1) and I need to find its solution. I apply the following steps, (y+1)/y*dy = x*sinx*dx 1 + 1/y*dy = x*sinx*dx ∫ 1 + 1/y*dy = ∫ x*sinx*dx ...skipping a few steps for convenience I get the equation y + lny = -x*cosx + sinx My problem...
  45. T

    A particular solution to the differential equation y'' + 2y' + y = t^2 + 3?

    A particular solution to the differential equation y'' + 2y' + y = t^2 + 3? is t^2 - 4t +7 in the answers, but i get t^2 - 4t + 9 so where am i going wrong... y = Ay^2 + by + c y' = 2Ay + B y'' = 2A 2A + 2(2Ay + B) + Ay^2 +By + c = t^2 + 3 A(y^2) + (4A+B)y +(2A +2B+c) = t^2 + 0t + 3...
  46. V

    What exactly is a 2nd order differential equation?

    A first order DE models the rate of change, e.g. when decay is proportional to time we have the DE: dM/dt = -K.M; this is describing that rate of change mathematically. Am I correct in saying that a 2nd order DE describes the rate of rate of change? Also, can anyone explain any application of...
  47. P

    Find the Solution to dy/dx = 3y-3y2

    Homework Statement Find the explicit solution of dy/dx = 3y-3y2 Be sure to include any singular solutions in your answer Homework Equations Not sure... The Attempt at a Solution dy/dx = 3y-3y2 dy/(3y-3y2)=dx ∫(1/(3y-3y2))=(1/3)logy-(1/3)log(1-y) ∫(dx) = x + c x+c =...
  48. B

    Orthogonality/infinite series solutions differential equation

    Homework Statement This is the problem statement in the picture for exersize 14, it's rather long (pertaining to orthogonality - which I only understand what the definition of orthogonality is, which is the "(15)" on the side of the image below. http://postimage.org/image/oxhw2uf8p/...
  49. A

    Differential Equation- Height of water in tank

    Homework Statement A water tank is filled by an inflow x(t), the tank is emptied by the outflow y(t) The outflow if controlled by a resistance R The water depth in the tank is represented by d(t) The surface area of the water is A, independent of depth The tank is 1.5m high with a...
  50. Y

    How to go about solving this first-order nonlinear differential equation?

    I saw this post at stackexchange: I ran across this post when trying to solve a homework problem. But I have no idea how he got that solution for that. When I use the Euler-Lagrange, I get this diff eq below. Here is the simplest form I have managed to get it in...
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