Differential equations Definition and 999 Threads

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

    First year calculus differential equations question

    Homework Statement Suppose \alpha satisfies 1) a_{n} \alpha^{n} + .. + a_{0} = 0 Suppose that polynomial splits into n real linear factors ( it has n real roots ) Then, any solution of a_{n} y^{n} + .. + a_{1}y' + a_{0} = 0 is of the form y(x ) = c_{1}e^{\alpha_{1}x} + ...
  2. C

    Finding the general solution of a system of differential equations

    Homework Statement The question is: Homework Equations I really don't know what to put here but my method is: -Find det(A-\lambdaI) -Find the roots of the determinant - which are the eigenvalues -Solve for (X -(\lambda)I) I am stuck at this part The Attempt at a Solution...
  3. S

    Usual convention for proving differential equations

    Hi there PF How does mathematicians usually prove differential equation (I am just speaking of the ordinary differential equations). We are going through proofs of differential equations in my high school math class at the moment, and in our books there are usual proofs, where one derives...
  4. X

    System of Linear Differential equations

    (I do not know how to make matricies with LateX, so where the semi-colon is denotes a new row!) t \vec{x'}=[-4,2;2,-1]\vec{x} The part that throws me off with this problem is the x' vector being multiplied by t! I know if the t wasn't there, all I would need to solve is det(A-(lamda)I)=0
  5. B

    Why Does the Wronskian of Two Functions Yield Zero?

    y1(t) = t^(2)+5t, y2(t) = t^(2)-5t I know that these functions are linearly independent because they are not scalar multiples, but every time that i do the Wronskian i get 0. [t^(2) + 5t t^(2)-5t] [2t+5 2t -5] (2t^(3) -25t) - (2t^3-25t) = 0 my instructor some how...
  6. M

    Solve Modeling Higher-Order Diff Eqn: Free Undamped Motion w/ Spring

    A mass of 1 slug is suspended from a spring whose spring constant is 9lb/ft. The mass is initially released from a point 1 foot above the equilibrium position with an upward velocity of square root 3 ft/s. Find the times at which the mass is heading downward at a velocity of 3 ft/s. I found...
  7. D

    System of coupled second order differential equations.

    Hey folks I'm looking for a way to find the characteristic equation for a second order coupled system of differential equations such as... \ddot{x} + A\dot{y} + Bx = 0 \ddot{y} + C\dot{x} + Dy = 0 Where x and y are functions of time. I know I can solve it by setting x and y to standard...
  8. P

    Doing Differential Equations For a Living

    So basically I am a pure math major intending on going to graduate school to get a phd in some area of pure mathematics (possibly something like algebra or functional analysis), and I want to be well prepared for the more than likely outcome that I won't be able to obtain a permanent job in...
  9. J

    Intro to Differential Equations Problem #1

    Homework Statement 1. Find the general solution to the differential equation: y''-5y'+6y = 0 Homework Equations The Attempt at a Solution m^2 - 5m + 6 = 0 (m-1)(m-5) = 0 y' = Ae^x+Be^5x
  10. D

    How are Differential Equations usually found in real world applications?

    I just started learning about DE's and am impressed by how much information they contain. Are they usually just a relationship between several simpler functions? For instance, a basic kinematic problem like v = at + vo seems quite hard to come up with experimentally but v = d/t would be...
  11. S

    Solving linear differential equations

    Homework Statement the full question is asking me to solve dy/dx + (1/x)y = 3cos2x i think i know what i am doing up to a point, but for me to continue with the question i have to integrate exp(x^-1)3cos2x and I am not sure how to do this, once i get this part i would know how to...
  12. T

    Iterative methods for solving systems of linear differential equations

    Hi. Anyway, learning this sort of makes me feel like I've chosen the wrong school for myself, but I'd like to try and see if I can understand this nevertheless. See, I'm supposed to learn various numerical methods for solving systems of linear equations, like the Jacobi or the Gauss-Seidel...
  13. P

    What is the Solution to a Differential Equation with Initial Conditions?

    Homework Statement (2x*y-5)dx+(x^2+y^2)dy=0 test for exactness and solve Homework Equations The Attempt at a Solution I've been following an example to try and solve this but I'm not sure if its right or how to finish it of. \delta M/\delta y=2x \delta N/\delta x=2x...
  14. K

    Homogenous constant coefficient linear differential equations

    Hello I am kinda confused when it comes to finding a general solution of equation. Here is the question. Y'' + 6Y' +9Y = 0 Solution Y'' + 6Y' +9Y = 0 I used "x" instead of Lamda...ANYWAY X^2 + 6X + 9 = 0 FACTOR IT WHICH IS (X+3) (X+3) = 0 X IS EQUAL TO -3. HERE IS WHERE I...
  15. N

    Differential Equations and Fourier Series

    Homework Statement Q1) (dy/dx)= 2x(y2+9); y(0)=0 Q2) (x4+y2) dx - xy dy =0; y(2)=1 Q3) (dy/dt)= 4y+t Q4) y"+2y'+y=0; y(0)=4 and y'(0)=-6 Q5) y"+3y'+2y= 30e2t 3. Solutions found A1) y= tan (x2/3) A2) this is not an exact differential and so cannot be solved A3) couldn't solve this...
  16. G

    Differential equations in Quantum Mechanics

    Studying a maths degree, going onto final year next year, am planning to do a 3rd year course in quantum mechanics. I just want to ask, how much probability theory and differential equations are there in quantum mechanics? Someone said that ultimately quantum mechanics is about probability...
  17. K

    Differential equations using Laplace Transform

    Homework Statement ((d^2)x)/(d(t^2)) + 5(dx/dt) + 6x = f(t) f(t)=3 for (0<=t<6) f(t)=0 for t>=6 Homework Equations This I am not sure of, this is my question. Would I use the Second Order Differential equation to do this problem: L{((d^2)f)/(d(t^2))} = (s^2)F(s) - sf(0") -...
  18. R

    Differential Equations: Compartment Model, when I = 0

    Homework Statement The pollutants are being added to the lake at a constant rate I, and the pollutants are thoroughly mixed into the lake. The annual precipitation into the lake matches evaporation, so the flow rate F is also constant. Let y(t) be the amount of pollutant in the lake and c(t)...
  19. S

    Applications of Differential equations

    The rate of capital growth in a bank account is described by the differential equation dM/dt = aM Where dM/dt is the rate of change of the capital M and a is the annual interest rate. Show that the general solution for the time dependence of capital M(t) is given by: M(t) = M0 * e^at...
  20. PainterGuy

    Books on linear algebra and differential equations

    hi, first let me inform you that I'm not very good maths but love to learn it. I've basic understanding of maths. u take my maths abilities as 9th or 10th grader high schooler. i don't exactly know what linear algebra is but someone told me that it's about matrices. i have 4x4 and 6x6...
  21. R

    Find equilibrium points given 2 differential equations

    Homework Statement Given a system of differential equations and asked to find the equilibrium points and classify them. Homework Equations Equation 1 ... dx/dt=y(13-x^2-y^2 ) Equation 2 ... dy/dt=12-x(13-x^2-y^2 ) The Attempt at a Solution I know the solution...
  22. S

    Coupled differential equations with convolution and correlation

    Homework Statement I have two equations: \frac{\partial}{\partial z}E_{L}\left(z,\omega \right) = i \frac{2 \mu d_{eff}(\omega+\omega_{0})^{2}}{k(\omega+\omega_{0})}\int E_{L}(z,\omega-\omega_{T})E_{T}(z,\omega_{T})d\omega_{T} \frac{\partial}{\partial z}E_{T}\left(z,\omega_{T} \right) = i...
  23. G

    Heat Conduction of Metal Rod (Differential Equations)

    Hi everyone, I recently started studying heat conduction using differential equations and this has been stumping me for a while. I am having trouble understanding what type of heat conduction problem this is. We are given a 100cm long copper rod with ends maintained at 0 C. The center of the...
  24. H

    The connection between variational principle and differential equations

    It is very well known that the result of varying some functional gives a differential equation which solutions minimizes the given functional. What about the other way around? Can one find a functional that is minimized given a differential equation? Is there a procedure for this? The reason...
  25. M

    On differential equations texts

    I have a feeling this is the wrong place to post this. I can never figure out where to put things :confused: Anyway, my professor uses the 4th edition of Edwards and Penney Differential Equations and Boundary Value Problems (https://www.amazon.com/gp/product/0131561073/?tag=pfamazon01-20)...
  26. C

    Classify fixed points non homogeneous system of linear differential equations

    Homework Statement \dot{x}=2x+5y+1, \dot{y}=-x+3y-4 Homework Equations The Attempt at a Solution Well, if system was: \dot{x}=2x+5y, \dot{y}=-x+3y we let a=2, b=5, c=-1, d=3. Then p = a + d and q =ad - bc and we investigate p^2-4q Don't know what to do when \dot{x}=2x+5y+1...
  27. O

    Coupled first order differential equations

    I am trying to solve a problem (not homework, too old for that! lol!) which involves the time dependent schrodinger equation for magnetic moment in time-dependent magnetic fields. I end up with the following that needs to be solved: x' = -i*(b*t-a*t^2)*x - i*c*y y' = -i*c*x -...
  28. C

    Initial Value Differential Equations problem

    dy/dt = 2y-5, y(0) = y_0 The answer I got for this was y =5/2 + (y_0-5/2)e^t but the solution is y =5/2 + (y_0-5/2)e^-2t Where did this -2t come from? My steps: dy = (2y-5)dt Integrating both sides gives 1/2*ln|2y - 5| = t + C take exp of both sides gives 1/2(2y-5) = Cet...
  29. T

    Differential Equations Trouble

    Homework Statement Verify that each given function is a solution of the differential equation. 1. ty' - y = t^2 ; y = 3t + t^2 2. y'' + y = sect , 0<t<pi/2 ; y = (cost)ln( cost ) + tsint The Attempt at a Solution int (tdy) = int(t^2 + y)dt which isn't y=3t + t^2...
  30. B

    Differential Equations and population

    Homework Statement Consider a population p of field mice that grows at a rate proportional to the current population, so that dp/dt=rp. Find the rate constant r if the population doubles in 30 days. Find r if the population doubles in N days. Homework Equations ...The Attempt at a Solution...
  31. K

    Elementary differential equations

    Homework Statement I can't solve the following problems: 38) Solve y' =y + et from yo =0 by Method 2, where the deposit eT at time T is multiplied by et-T. The total output at time t is y(t) = (Integral)(eT * et-TdT) (The integral goes from 0 to t). Substitute back to check y' =y + et...
  32. M

    Differential Equations problem 2

    Homework Statement 2(2x^2+y^2)dx-xydy=0 Homework Equations let x = yv dx = ydv+vdy The Attempt at a Solution (4v^2y^2+2y^2)(ydv+vdy)-vy^2dy=0 4v^2y^3dv+4v^3y^2dy+2y^3dv+2vy^2dy-vy^2dy=0 4v^2y^3dv+2y^3dv+4v^3y^2dy+vy^2dy=0 2y^3(2v^2+1)dv+y^2(4v^3+v)dy=0 dividing all sides by...
  33. P

    Engineering Series RLC Circuit & Differential Equations

    Homework Statement I have a series RLC circuit, no values given, connected to a voltage source Vs(t). I am asked to write the differential equations for: a. that relates the inductor current iL(t) to the source voltage Vs(t). b. that relates the capactor voltage Vc(t) to the source...
  34. S

    2 methods to solve system of differential equations

    Homework Statement Greetings – I am wanting to solve the following system of differential equations: K*\frac{d^2x}{dt^2} = -K2*\frac{dz}{dt} K*\frac{dz^2}{dt^2} = K2*\frac{dx}{dt} Homework Equations The Attempt at a Solution Now – I am not just...
  35. T

    Clearing the path to Differential Equations

    According to my school, I got to take calc 3 and 1 semester of linear algebra before i can start differential equations. Is this so? Anything else you would advise to add? Or is this too much and I can deal with less? I ask because many have advised calc 3 before any physics even though my...
  36. A

    Homogeneous linear differential equations

    Homework Statement The equation 2y'' - y' + y2(1 - y) = 0; where y' = dy/dx and y'' = d^2y/dx^2 represents a special case of an equation used as a model for nerve conduction, and describes the shape of a wave of electrical activity transmitted along a nerve fi bre. Homework...
  37. K

    Introductory Differential Equations Question

    Thank you everyone for being a source of help in previous problems I've posted here. I'm starting an intermediate course in Differential Equations and I'm enjoying it so far, but this one problem on my homework seems to be giving me a problem and I think that I haven't fully grasped the...
  38. R

    Conditions for Exact Differential Equations

    Homework Statement Let M(x,y) = yf(xy) and N(x,y) = xg(xy), where f(v) and g(v) are functions of a single real variable v, defined and continuously differentiable for all real values of v. Under what conditions on f and g is the form Mdx + Ndy exact for all values of x, y in the plane? In that...
  39. H

    What are some good intro to differential equations books?

    My professor has us using C. Henry Edwards and David E. Penney, Differential Equation and Boundary Value Problems, Fourth Edition. However, I don't really like it as our textbook. Anyone else have any recommendations?
  40. K

    Power series solutions to differential equations

    Homework Statement I'm revising at the moment and a bit stumped on question 4 http://www.maths.ox.ac.uk/system/files/attachments/AC104.pdf Homework Equations The Attempt at a Solution I think for the first part of the question, the regular singular points are 0 and -2...
  41. F

    Nonlinear Differential Equation Homework Attempt

    Homework Statement r" = 1/r^2 Homework Equations A friend gave this to me, he was just wondering how we'd approach it. The Attempt at a Solution I don't think the equation is linear, so I don't know how to approach it. My friend suggested integrating both sides with respect to r...
  42. A

    Differential Equations - 1 vs 2 Variables?

    greetings, is it true that a exact differential equation is for two independent variable whereas linear differential equation is for one variable? advanced thanks,
  43. Rasalhague

    Differential Equations Text: Finding Rigor & Clarity

    Can anyone recommend an elementary but rigorous text on differential equations that takes care to distinguish between a function, f: X \rightarrow Y, and its value f(x) \in Y, as Spivak does in his Calculus. I find I'm getting rather confused by the traditional y=y(x) type dual-meaning notation.
  44. T

    Linearizing ordinary differential equations

    Hi guys, I am trying to linearize a coupled non-linear ode. I used partial derivative, and then jacobian matrix, i have seen paper using state-space model of jacobian matrix. I can't get a proper reference on this state-space model. Attached is the non-linear ode, the partial derivative of the...
  45. J

    Differential equations- simultaneous equations

    Homework Statement http://img228.imageshack.us/i/cimg5162f.jpg/ This isn't the usual format, so I hope it's still ok. It's just easier than writing it out on here. I can't work out how to form a simultaneous equation from (1) and (2). When solved it should form the du/dx and dv/dx...
  46. D

    Self Teaching of Differential Equations

    By all techinical standards, I am not required to take Differential Equations. I am currently a Physics major with a 2nd major in Mathematics Education. I want to take DE before I start any of my upper level Physics courses next fall since 2 of my academic advisors (one for Physics, one for...
  47. O

    Nonhomogeneous second order linear differential equations

    Homework Statement i'm supposed to find the general solution of the equation: y'' + 3y' + 2y = e^x + e^-x Homework Equations I have no problem with solving this equation however, i am confused with the step they are taking in the solutions (circled)...
  48. N

    Solving Second Order Differential Equations using Runge Kutta

    Homework Statement In aerodynamics, one encounters the following initial value problem for Airy’s equations: y''(x) + xy = 0, y(0) = 1, y'(0) = 0 Using the Runge-Kutta method with h=0.005 and determine values between x=0 and x=10 sufficient to sketch the relationship...
  49. S

    Higher order differential equations and the chain rule (2 variables)

    Homework Statement The function F is defined by F (r, θ) = f (x(r, θ), y(r, θ)), where f is twice continuously differentiable and x(r, θ) = r cos θ, y(r, θ) = r sin θ. Use the chain rule to find d2F/dθ2Homework Equations The Attempt at a Solution I know that dF/dθ = (df/dx)(dx/dθ) +...
  50. H

    Linear Algebra or Differential equations?

    Which one would you recommend me to take over the summer? Or should I take both? Has anyone ever done that? Next semester I will be taking multivariable calculus and i just wanted to take a math class over the summer because I really enjoy it. Each class is 3 hours long and meets MTWTR.
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