What is Diff eq: Definition and 271 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. K

    Question about a Riccati type diff eq.

    \acute{y} = \frac{1}{t^{2}} -\frac{y}{t} - y^{2} To solve a Riccati type diff eq a particular solution is needed. In our case it is: y_{1} = -\frac{1}{t} By setting y= y_{1}+\frac{1}{u}= -\frac{1}{t}+\frac{1}{u} and differentiating this \acute{y}= \frac{1}{t^{2}}-\frac{u'}{u^{2}}...
  2. M

    Falling Raindrop - Evaporation and - Diff Eq project

    I'm working on a project where I'm trying to find the differential equation of a falling raindrop. The drop I'm considering will be under 10mm to eliminate the drops breaking apart and I've found that for that size, the drops will be spherical and traveling 6-8m/s. I've determined my air...
  3. C

    Diff eq, power series solns, how do i determine how many terms to pull?

    Homework Statement find 2 power series solutions of the given diff eq about the ordinary point x = 0 y'' - xy = 0 Homework Equations y = (c_0)(y_1)[x] + (c_1)(y_2)[x] The Attempt at a Solution i can set it up to this (sorry idk out how to insert the subscripts with the summation symbols)...
  4. D

    Differential Equations: Seeking Advice to Make It Interesting

    Hey all, So I need some advice on my differential equations course. Last semester I took Calc 3 and loved it. I loved that I could visualize everything. It made math seem so much more concrete. On the other hand, there is differential equations. Now I know that there are an immense amount of...
  5. H

    What is the Differential Equation for Velocity with Air Resistance in Free-Fall?

    Homework Statement An object in free-fall has an acceleration (which is the rate of change in velocity with respect to time) of 32.2 feet per second2 downward plus air resistance. The air resistance is proportional to the velocity squared. If the initial velocity is 0 feet per second and the...
  6. I

    Diff EQ - Heating and cooling w/ fluctuating outside temp

    Homework Statement Say we have a warehouse with a 5 hr time constant. The outside temp is 16C at 2:00AM, and 32C at 2:00PM. The warehouse is at 24C at noon. What will the temp be at 6:00 pm? When will the temp be 27C? The Attempt at a Solution First, I set up a sine wave for the...
  7. I

    Diff Eq tank problem not working out

    I know you guys must get tired of the tank problem, but I just can't seem to get this figured out. This is my first post here, so hopefully this is in the right subforum and is presented properly. "A brine solution of salt flows at a constant rate of 4L/min into a large tank that initially...
  8. H

    Diff Eq Solution for E&M Example: Solving y'' and z'' with Author's Method

    Homework Statement This is part of a solution to a E&M example from a book. I am in the middle of following the solution and I come across this system of diff eq's:y''=\omega z' z''=\omega (\frac{E}{B}-y')Homework Equations Solution according to author is: y(t)=C_{1}\cos (\omega t)+C_{2}\sin...
  9. C

    Diff Eq: Equilibrium Solution sketching?

    Homework Statement \frac{dy}{dt}=r(1-\frac{y}{K})y Homework Equations The Attempt at a Solution 0=r(1-\frac{y}{K})y y=0 and K. Plotting dy/dt vs y, the intercepts would then be (0,0) and (0,K). The book says "the vertex of the parabola is (K/2, rK/4)." Is this something...
  10. H

    Could you help me solve first order diff eq?

    dv/dt=g-(k/m)*v^2 i thought about all sorts of diff eq solving techniques but nothing comes up to my mind dv/(g-(k/m)*v^2)=dt but it doesn't really help ...
  11. D

    How Do You Plot Solutions and Understand Coefficients in Differential Equations?

    Homework Statement Solve each of the following initial value problems and plot solutions for several values of y_0. a) dy/dt=-y+5, y(0)=y_0 b) dy/dt=-2y+5, y(0)=y_0 c) dy/dt=-2y+10, y(0)=y_0 Homework Equations - The Attempt at a Solution I have solved all of the equations for...
  12. B

    How Do You Model Drug Absorption in a Patient Using Differential Equations?

    Fluid is entering a patient at a rate r, given in cm^3/hr, and a drug is present at some amount of mg/cm^3, m. The drug is absorbed or leaves at a rate given in h^(-1), c, proportional to the amount of the drug. I want to write a differential eq describing amount, a, in terms of t. I...
  13. K

    Laplace Transform Diff EQ Help Request

    Ok...so this isn't homework (well past that at 39!) but I'm trying to brush off the old cobwebs and solve a vehicle motion problem. Specically, I'm trying to find the Homework Statement Ok...so this isn't homework (well past that at 39!) but I'm trying to brush off the old cobwebs and...
  14. F

    Diff eq Fourier Series Stuck halfway through,

    Homework Statement Find the Fourier series of f on the given interval: EDIT: For the result of an, it should be multiplied by \frac{2}{1+n^{2}}. That has been corrected.
  15. E

    Simple Diff EQ - Modeling the salt concentration of a pond

    Homework Statement The problem can be found here: http://alex.fangsoft.net/math.pdf I have completed Task #2, it is Task #1 that I am very lost on. Our teacher has really thrown us head first into the water with this one, I am very unfamiliar with differential equations and their...
  16. F

    Diff Eq. Easy question on prob. Please help last step

    Diff Eq. Easy question on prob. Please help...last step Homework Statement Solve The Attempt at a Solution So this is my attempt. I just need to figure out how to adjust Y(s) so I can take inverse laplace transform. I have made the denominator a single fraction, whose denominator is s2+1 and...
  17. H

    Modeling with first order Diff Eq.

    Homework Statement The initial mass of fish in a lake was 7 thousand pounds on January 1st, 2001. Since the time, there was a 4-year moratorium on the harvesting on this specific type of fish. This species of fish reproduce at a rate proportional to the mass and by next year on the same...
  18. Saladsamurai

    What is the type of DE and solving method for y'' + (x^m)y = 0?

    Homework Statement I have the following DE: y'' + x^my = 0Homework Equations Maybe Frobenius or Power series or Bessel.The Attempt at a Solution I am having a hard time nailing down the type of DE this is. Clearly it is 2nd order homogeneous with variable coefficients. If m is a *...
  19. A

    Can Differential Equations Be Used to Create Art?

    Hello all, I have been assigned a project to create artwork using differential equations, the obvious idea would be to use Mathematica or Maple to obtain general solutions and replace values to obtain different level curves and then fill in the space between them. However, I'm wanting to...
  20. Saladsamurai

    Solve Homogenous Diff EQ: x\sqrt{x^2 - y^2}\,dx = (x+y)(y\,dx - x\,dy)

    Homework Statement I was given: x\sqrt{x^2 - y^2}\,dx = (x+y)(y\,dx - x\,dy)\qquad(1) \Rightarrow x\sqrt{x^2 - y^2}\,dx = xy\,dx - x^2\,dy +y^2\,dx - xy\,dy Dividing by dx we have x\sqrt{x^2 - y^2} = xy - x^2\frac{dy}{dx} +y^2 - xy\frac{dy}{dx} \Rightarrow x\sqrt{x^2 - y^2} -xy - y^2 =...
  21. J

    Math (Subject) GRE without Diff EQ

    I just realized I won't have the chance to take a class in differential equations before I have to submit my math graduate school applications, so there's no way I'll have the class beforehand. I've taken pretty much every other topic mentioned, and will review heavily on the calculus (it's been...
  22. F

    Diff eq, Spring/mass damped/driven-Am I doing this right?

    Diff eq, Spring/mass damped/driven--Am I doing this right?! Homework Statement 1.) After a 10lb weight is attached to a 5ft spring, the spring measures 7ft. The 10lb is replaced with 8lb, and placed in a medium offering resistence equal to the instantaneous velocity. A) find the equation of...
  23. F

    Non-Linear DiffEq: Solve y''+(y')2+1=0

    Homework Statement Solve the given differential equation by using the substitution u=y' y''+(y')2+1=0 The Attempt at a Solution If I got to this last integral correctly, I don't know how to solve it. I'm thinking I didn't get to that step correctly.. But I can't find my mistake..
  24. F

    Diff eq: I dont understand my prof's comments. Here is my work.

    So I had an exam and got this problem wrong, that I had done incorrectly on the quiz as well. comments in red under picture are comments from professor. here is the work: Attempt) Why is this general solution? What is the fundamental set? What is yp? (this should be verified) So on the...
  25. S

    Linear systems diff eq question

    given x(t)=impulse find y(t) y(prime)=4y=8x i am lost
  26. Saladsamurai

    Diff EQ: Method of Undetermined Coefficients

    Homework Statement Solve using Method of Undetermined Coefficients: y'' - 2y' + 3y = 15t - 8 The Attempt at a Solution First I used the characteristic equation to solve for the homogeneous solution to get: yh = C1e-t + C2e3t Great. Now I need to find the particular. I am...
  27. N

    Diff eq problem appears simple but is deadly

    [b]1. Given i. dx/dt + y = 0; x(0) = 1 : ii. dy/dt + x = 0; y(0) = 0 Solve for the solns of x and y Although these are first order dif eq's I worked them as leading to a second order undetermined coefficients problem First I wrote them as i. x' + y = 0 and ii. y' + x = 0 Eqn. ii...
  28. Saladsamurai

    Diff Eq: Is it me? Or Wolfram? I know, it's me.

    Homework Statement y(5) - 2y(4) + y(3) = 0 This is 5th-order, linear, HG, with constant coefficients, and hence I used the characteristic equation: m5 - 2m4 + m3 = m3(m - 1)(m - 1) = 0 Hence, m = {1,1,0,0,0} and so the solution is given by: y(x) = C1 + C2*x +C3*x2 +...
  29. K

    Help to stregnthen laplace and second order diff eq beckground

    Homework Statement Here is my issue. I am an EE student that has been part time for almost five years. I took diff eq and signals a long time ago. I am currently taking a feedback and controls class, and my lack of memory in these topics is killing my study time. I am looking for relevant...
  30. M

    Rocket Propulsion Differential Equation Solution Help?

    Homework Statement Consider the case of a rocket taking off vertically from rest in a gravitational field g. The differential equation is given by m\dot{v} = -\dot{m}v_{ex}-mg Assume the rocket ejects mass at a constant rate, \dot{m}=-k (where k is a positive constant), so that m=m_{0}-kt...
  31. F

    Diff eq, need a simple step explained

    So its not a problem, but a step in an example that I need explained. The section is solutions by substitutions and its an example of solving a homogeneous DE. Step goes from: \left(x^{2}+u^{2}x^{2}\right)dx+\left(x^{2} - ux^{2}\right)\left[udx + xdu\right]=0 to: x^{2}\left(1 + u\right)dx +...
  32. J

    Diff EQ Intro - Verify Family of Functions as Solution

    Homework Statement Verify that the indicated family of functions is a solution of the given differential equation. Assume an appropriate interval I of definition for each solution. Homework Equations dy/dx + 2xy = 1; y = e-x2\int(from 0 to x)et2dt + c1e-x2 The Attempt at a Solution...
  33. A

    Solving First Order Diff EQ: Salt Tank Example

    Homework Statement Salt water pours into a 10 liter tank at a rate of 4 l/min. Its concentration is 2 g/l. The brine in the tank is well mixed and it drains out at a rate of 4 l/min. Call y the grams of salt in the tank at time t. The tank is initially full of fresh water. Solve the...
  34. H

    Diff eq. and linear algebra concurrently and before calc III?

    So my engineering program has me taking LA and Diff. Eq. in the same semester, after Calc II and before Calc III. I don't understand why that is...what would be the idea behind this?
  35. A

    1st order linear diff eq. problem

    Homework Statement Find the general solution of \frac{dy}{dt} = 2y +sin(2t) Homework Equations The general solution of a nonhomogeneous ode is the particular solution of the nonhomo plus the solution of the homogeneous ode.: y(t)= y_{p}(t)+y_{h}(t) The Attempt at a Solution...
  36. T

    Having trouble solving this diff eq

    So I am an engineer and its been about two years since I have taken a differential equations class and the method in which to solve this has totally been lost to me. If anyone can help me out that would be great thanks. m1x1''+k(x2-x1)+b(x2'-x1')+F(t)=0 m2x2''+k(x1-x2)+b(x1'-x2')=0 I...
  37. jinksys

    Find a general solution [Diff Eq]

    Homework Statement Find a general solution for y''' - 6y'' + 9y' = 0 Homework Equations The Attempt at a Solution I know that the general solution for a homogeneous DEQ is Y(x) = c1y1(x) + c2y2(x) ... cnyn(x) however, I am not given y1, y2 , or y3 so I am to assume that the...
  38. C

    Is it a bad idea to take Calc III and Diff EQ in the same Semester?

    Is it?
  39. F

    Solve Linear Systems: Determine Critical Point Stability

    Wanted to brush up on my diff equations a bit. I have kind of forgotten how to determine phase plane stability in systems of differential equations. Homework Statement Solve linear systems. Determine whether the critical point (0,0) is stable, asymptotically stable, or unstable. dx/dt=2x...
  40. S

    Supplements Apostol 2nd Edition: Linear Algebra and Diff EQ

    Has anyone used Apostol's 2nd edition textbooks for Calculus? They contain lots of information about linear algebra and differential equations. But do they contain enough to be equivalent to a first and/or second course in them? If they do not, what supplementary texts do you recommend to...
  41. A

    Ordinary Diff Eq. - Help with Calculus

    Homework Statement Hey all. I'm having issues with a few Diff. Eq. problems. I think I'm screwing up at the point of u-substitution, but I do not know for sure. Homework Equations I'm evaluating: (x2 + x3)*integral (x4/(x4+2x5+x6) The Attempt at a Solution I eliminate x4 inside the...
  42. B

    Newton's Law of Cooling (diff eq. - seperation of variables)

    Newton's Law of Cooling (diff eq. -- separation of variables) Homework Statement Fresh coffee sitting in a room cooling...you know the routine. Anyhow T(0) = 90degreesCelcius. Room temp=25degrees Celcius find k. Then he asks us to use Euler's method to estimate coffee temp after five...
  43. L

    Checking whether my Diff EQ limit problem is correct

    figured it out, thanks Figured it out, thanks.
  44. D

    Laplace Transform of Bessel Diff Eq

    Hello PF, maybe you can help with this one! I need to show that the Laplace transform of J0(at) is (s^2 + a^2)^-1/2. My prof told me to start with the form: x2y'' + x y' + (x2 + p2)y = 0, where p = 0 ITC. What have I got so far?... Doing the Laplace transform on both sides, where Y...
  45. T

    Confusion - Diff Eq for a Circuit? What does my professor mean?

    Homework Statement My professor wants us to derive a differential equation for the current in a circuit. The circuit has a battery with voltage E, resistor with resistance (iR)??, and inductance L. It's in series. The Attempt at a Solution What REALLY confuses me are these things...
  46. J

    2nd order diff eq, 3 dim. and wierd boundry condition

    I need help. For the following problem, can someone suggest how I should start on this question. I only have one quarter of diff eq classes plus a few classes in Fourier analysis. I'm out of my league. Consider a box with length, width and height given by L. The box encompasses the region...
  47. O

    Inhomogeneous diff EQ, undetermined coefficients

    Find the solution of: y" + 3y' = 72sin(3t) + 36cos(3t) where y(0) = 6 and y'(0) = 9 I first found the solution to the homogeneous eq: the roots (R^2 + 3R = 0) are R = 0, -3 which gives the family of solutions: y = a(1) + be^(-3t) and y' = -3be^(-3t) using the initial conditions...
  48. P

    Solving a 1st-Order Diff Eq for Boating Path

    Homework Statement The following problem involve the setting up and solving of a first-order differential equation for a physical situation. Once, derived the equation itself is not difficult to solve. A boater rows across a straight river of constant width "w", always heading (i.e., pointing...
  49. Saladsamurai

    How to Solve a Differential Equation Using Laplace Transforms?

    Homework Statement Given: dx/dt + 3*x = exp(-3*t) and all initial conditions are zero. Homework Equations Laplace The Attempt at a Solution L[dx/dt + 3*x = exp(-3*t)] s*X(s) + 3*X(s) = 1 / (s + 3) X(s) = 1 / (s + 3)^2 So here is where I get mixed up. For some reason, I thought I was...
  50. O

    Best Differential Equations Books for Beginners: Recommendations and Reviews

    I was wondering if anybody could refer me to a few good differential equations books? I took an intro to diff eq class, but due to my job changing my work hours to the same time as the class I didn't get the full experience of the class (i got a B, but I don't really understand the material it...
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