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. M

    Elementary differential equations: Linear equations

    Homework Statement dr/d∅+rtan∅=cos∅ μ(∅)=exp[∫tan∅] μ∅=exp[-ln[cos∅]=-cos∅ -cos∅(dr/d∅)-rtan∅cos∅=-cos^2∅ dr/d∅[-rcos∅]=-cos^2∅ rcos∅=∫cos^2∅ changed cos^2∅ to 1/2+1/2cos2∅ rcos∅=1/2∅ + 1/4 sin2∅+c the books answer is not even close to mine they have r=(∅+c)cos∅...
  2. M

    Differential Equations Question

    Homework Statement Match each of the differential equations with a solution from the list below. 1. 2x^2 * y'' + 3xy' = y 2. y'' + 12y' + 36y = 0 3. y'' - 12y' + 36y = 0 4. y'' + y = 0 And then the possible solutions A-D Homework Equations The Attempt at a Solution This is...
  3. S

    Self-Study Differential Equations: Textbook for EE Major

    Unfortunately I am unable to fit a first course in differential equations into my schedule, therefore I have opted to self study the said course. I am looking for a textbook appropriate for an introduction into differential equations that isn't too ambiguous to tackle on my own. I am an EE major...
  4. M

    Elementary Differential equations : seperable making it explicit

    Homework Statement (y+2)dx +y(x+4)dy=0 Homework Equations The Attempt at a Solution ∫(1/x+4)dx + ∫(y/y+2)dy=0 ln(x+4)-2ln(y+2)+y=lnc Here is where I get confused how do I make this into an explicit solution, that +y really bothers me. I was thinking ln [...
  5. M

    Elementary Differential equations : seperable

    Homework Statement Tan∅ dr + 2r d∅=0 Homework Equations The Attempt at a Solution ∫1/2r dr +∫1/tan∅ d∅=0 1/2ln(2r)+ln(tan∅)=c ln[r(tan∅)]=ln(c) r(tan∅)=c the solution in the book says the answer is rsin^2∅=c Where did I go wrong? Thank you
  6. G

    Differential Equations. Solve in two ways.

    Homework Statement Solve y''=1+(y')2 in two ways, for x is missing and y is missing. Homework Equations Integration, and reduction of order. The Attempt at a Solution First method: (this is correct) y''=1+(y')2 let y'=p and y''=p' p'=1+p2 p'/(1+p2)=1 ∫p'/(1+p2)dx=∫1dx arctan(p)=x+c solve...
  7. G

    Differential Equations. Reduction of Order. Calc IV.

    Homework Statement (x2+2y')y''+2xy'=0 with initial conditions of y(0)=1 and y'(0)=0 Homework Equations reduction of order, and exact equations.. The Attempt at a Solution let y'=p and y''=p' Giving: (x2+2p)p'+2xp Let M=2xp and N=(x2+2p) So, Mp=2x and Nx=2x Therefore, they are...
  8. T

    Differential Equations - Help with scaling / dimensionalizing

    Hi all, I have a homework assignment related to the conservation of energy in a fluid. This is given in terms of the density ρ, specific heat c_{p}, thermal conductivity k, bulk viscosity μ, dissipation function \widetilde{\phi}_{v}, velocity function \widetilde{u}, spatial derivative of the...
  9. D

    Modeling with first order linear differential equations

    Homework Statement uploaded file (Problem 5)Homework Equations uploaded fileThe Attempt at a Solution Ok so I think I have the correct answer but the problem is I don't know how to solve for the actual answer with 2 equations which one do I plug into? Sorry for all the erasing!
  10. N

    Mathematica Mathematica: Coupled differential equations

    Hi I am trying so solve a set of coupled ODE's. I have been able to get a solution from Mathematica using NDSolve, but when I change the coefficients an error appears: "NDSolve::mxst: Maximum number of 10000 steps reached at the point t == 0.0000748339914040257`. >>" I have tried...
  11. S

    How to solve the attached differential equations using Mathematica

    Hello everyone, I am making use of Mathematica to solve differential equations developed by Lagrange method, I have met some problems in solving the equations, I cannot obtain the correct result even though I have set "t" and "MaxSteps" both as "Infinity". I have run the program for 7...
  12. dkotschessaa

    Differential Equations Intro material

    Homework Statement What function do you know from calculus is such that its first derivative is a constant multiple k of itself? (Do not use the function f(x) = 0.) Homework Equations The previous question asked what function was a derivative of itself - with the pretty easy answer...
  13. S

    NDSolve problems in Mathematica (differential equations)

    Hello everyone, I am making use of Mathematica to solve differential equations developed by Lagrange method, I have met some problems in solving the equations, I cannot obtain the correct result even though I have set "t" and "MaxSteps" both as "Infinity". I have run the program for 7...
  14. 1

    Life cycle of butterfly Differential equations

    Would someone be able to help me with the question Life cycle of butterfly is egg -> caterpillar -> butterfly. Each week 5% of the eggs will hatch and 2% of the caterpillars will turn to butterflies. You start with 100 eggs and 60 caterpillars. Using calculus to obtain an equation...
  15. D

    Linear and non-linear differential equations

    I am not sure I understand Linear and non-linear differential equations properly so I will ask some question which someone will hopefully answer. Is dx/dt = x + 2 a linear differential equation? If so does this mean that the rate of change of is constant? Is dx/dt = x^2 + 4x a non linear...
  16. S

    What to review for differential equations?

    Hi. I'm not sure if this is the right place for this kind of question, but it's related to coursework. I took Calc III during senior year of high school and linear algebra during freshman first semester. Then, I haven't taken any math class because I tried to fill my prereqs out and also take...
  17. J

    Analytical solution for coupled partial differential equations

    Hello, In my study i came across to solve the analytical solution for coupled equation y(x,t) and z(x,t).The equations contains" f " function which is a function of the first variable exponentially. The first equation is : ∂y/∂t=∂^2(y)/∂x^2- 2*f(y)*z; The second equation ...
  18. M

    I know nothing about differential equations

    Hello I know nothing about differential equations, can you explain them to me in a few simple words? And what's so special so we study them seperately? Thanks
  19. A

    Resource for modelling with differential equations

    I'm looking for a good resource about modelling for differential equations. I've completed two courses on ODE's and PDE's, but they both lacked in the applications/modelling department. Can anyone recommend a good resource/book on modelling?
  20. M

    3 Differential Equations problems

    Hey Can I have your help with: 1. (2e^y - X)y'=1 y(0)=0 y(x)=? 2. xyy' + y^2 - x^2 = 0 3. x^2y' - 1 = cos2y lim y(x) = π/2 x->+∞ thanks
  21. A

    Coupled partial differential equations

    Hello every one, In my physics problem, i end up having two coupled second-order nonlinear differential equations where the coupling terms include, the variable, the first derivatives, and also a second derivative coupling. I appreciate any help on how to handle this system before setting it...
  22. B

    Non-linear differential equations

    Homework Statement Show that w(t) = tanh(t) solves the nonlinear problem: w''(t)+2w(t)-2w3(t) = 0 t ε ℝ Homework Equations \frac{d^2tanh(t)}{dt^2} = -2tanh(t)sech2(t) = \frac{-8sinh(2t)cosh^2(t)}{(cosh(2t)+1)^3} tanh(t) = \frac{sinh(2t)}{cosh(2t)+1} tanh(t)3 =...
  23. P

    Differential Equations Compartmental Analysis Word Problem

    Hello, I am trying to tackle a word problem in differential equations, but the setup has flummoxed me. Thank you in advance to any advice/help given. Homework Statement Beginning at time t  0, fresh water is pumped at the rate of 3 gal/min into a 60-gal tank initially filled with...
  24. Chris L T521

    MHB DE Tutorial - Chapter 1: The Basics of Ordinary Differential Equations

    Hello everybody! This is the first post for the differential equations tutorial. Unlike the one on MHF, I've decided to take this project a little more seriously and have taken on the task of making a book based on these posts (well, it's really the other way around - the posts will be the...
  25. H

    Interest in Linear algebra and Differential Equations?

    I'm a college student whose been force enrolled in Linear Algebra and Differential Equations. No offence to math lovers but I don't find them very interesting. I study much better in courses which I find interesting so could someone please point out interesting applications of L.A. and D.E...
  26. C

    Ordinary Differential Equations

    Homework Statement Find all eigenvalues and eigenfunctions: -y'' (x) = \lambda y(x) , x \in (a,b) y(a)= y(b) =0 . Homework Equations sin x = \frac{e^{ix} + e^{-ix} }{2i} . The Attempt at a Solution So actually the only problem I have is to find the eigenfunctions (which should...
  27. U

    System of piecewise differential equations - Where to start?

    Hello all, I am dealing with a rather complicated problem (as we all do), without much knowledge in differential equations. I have coded a numerical solution, taking a straightforward approach: constructing the functions step by step, but I would like to try to achieve better precision...
  28. N

    MATLAB Matlab - How to solve simultaneous non-linear differential equations?

    Matlab -- How to solve simultaneous non-linear differential equations? How to solve simultaneous non-linear differential equations in MATLAB m-file without using the commands?. can you give me the program?.
  29. K

    Okay I have a problem with using variables in differential equations

    Consider the differential equation where I''+I=e^it where i= sqrt (-1) So I don't know what to do. I do know that The second derivative of c*t*e^(i*t) is -c e^(i t) (-2 i+t) But I don't know what to do for a and b and I need help to get threw this problem before I take my test...
  30. H

    Can Differential Equations Like This Be Solved?

    This isn't a homework problem, but it's a problem I've come across that I would really like to know the answer to. Is it possible? Homework Statement Solve for u(x) and v(x) in the system of equations: u(x)v'(x) = x^x(ln(x) + 1) u'(x)v(x) = x^xln(x) 2. The attempt at a solution...
  31. A

    Deriving Probability Density Functions from Partial Differential Equations?

    Deriving Probability Density Functions from Partial Differential Equations? Hiyas, I have been told that it is quite normal to get PDFs (Probability Density Functions) from PDEs (Partial Differential Equations). That in PDEs that the function can be a PDF and you can get this by solving the...
  32. J

    Solve set of differential equations

    I am looking for the solution of C_{3.p}(t) Edit: sorry I forgot to add the explantions for the symbols as HallsofIvy pointed rightfully out. - There are n+m+p equations with n+m+p variables of C - A, B, C, D, E, n, m, p are constantsusing the following set of equations: Part I \frac...
  33. D

    Partial differential equations class with only calc III?

    And not taking ODE's? Is this doable? I understand the basics of most concepts as I am currently self-learning from online resources and textbooks, but I decided not take the class during the summer as I'm already taking calc III. The problem is that when the year starts up again PDE's is taught...
  34. J

    How are differential equations developed

    When differential equations are being developed, what dictates the order of the differential? What decides if it is second order, third order, fourth order, fifth order, etc... I understand that the process is taking the derivative of a derivative of a derivative, but what decides if the the...
  35. J

    Tanks in series model - Differential equations

    Dear, I work as a process engineer and haven't had to use differential equations or integrated anything for years. I have the following physical problem and please advice me if this is not the correct location on the forum. Believe me or not but this problem has been researched for over 20...
  36. C

    Euler-Bernoulli Beam Theory and Nonlinear Differential Equations

    I've been reading through my mechanics of materials textbook recently, notably in regard to the section on the deflection of beams. The well regarded Euler-Bernoulli beam theory relates the radius of curvature for the beam to the internal bending moment and flexural rigidity. However the theory...
  37. L

    Differential equations substitution problem

    Homework Statement Solve using an appropriate substitution ydx+(1+ye^x)dy=0 Homework Equations N/A The Attempt at a Solution u=y du/dx=dy/dx u+(1+ue^x)du/dx=0 This is where it gets really sticky for me. I can't see it being a separable variable problem because it seems...
  38. R

    Second Order Differential Equations

    Homework Statement Solve the following initial value problem: \frac{d^2y}{dt^2} + 2 \frac{dy}{dt}+y = t y(0) = 1, \ y'(0) = 0 The correct answer must be: y(t)=3e^{-t} + 2 t e^{-t} + t -2 The Attempt at a Solution Where did they get the term 2te-t from? Here's how I've done it so far...
  39. H

    Cramer's Rule application in differential equations

    We have this system of equations: \begin{cases} x'= -x + 2y & (1)\\ y' = -2x - y + e^{-t} & (2) \end{cases} where x(0) = 0 ; y(0) = 0 We apply the Laplace transform on (1) and (2) and get: (s + 1)X - 2Y = 0\\ 2X + (s + 1)Y = \frac{1}{s + 1} We can use elimination here...
  40. B

    Solving linear differential equations by factoring

    Solving linear differential equations by "factoring" I have thought of an interesting way of solving n-th order linear differential equations (with constant coefficients) by imitating the way we solve n-th order polynomials, that is by "factoring" it into a "product" of simple 1st order linear...
  41. K

    Differential Equations - Fick's Law

    Homework Statement A cell contains a solute at a concentration of Ct and the concentration of the same substance outside the cell is a constant, k. By Fick's law, if ct and k are unequal, solute moves across the cell wall at a rate proportional to the difference between ct and k, towards the...
  42. A

    Differential Equations? Fill in the table.

    Fill in the missing values in the table given if you know that dy/dt=0.2y. Assume the rate of growth given by dy/dt is approximately constant over each unit time interval and that the initial value of y is 6. Table: t-0-1-2-3-4 y-6-?-?-?-? (fill in the missing values where there is a...
  43. L

    Differential Equations 2 order and objectives

    Help pls decide this Differential Equations!I did not coincide with the answer and in some no idea how to do! will be very grateful if you start a proper solution to Us 1) y(4)+2*a2*y''+a4*y= cos(ax) 2) y' = x + 1/y; y(0)=1 3) y' = 2*x + cos(y); y(0) = 0 4) y(x) - λ*(0 to...
  44. T

    Coupled Nonlinear Differential Equations

    Hey, I need your help to solve the following set of coupled differential equations numerically. dn(t,z)/dt=I^5(t,z)+I(t,z)*n(t,z) dI(t,z)/dz=I^5(t,z)-α(n(t,z))*I(t,z) where I(t,0)=I0*exp(-4ln2(t/Δt)^2) and n(t,0)=0 and n(-certrain time,z)=0. Some constant parameters I did not show...
  45. L

    Differential Equations 2 order and objectives

    Help pls decide this Differential Equations!I did not coincide with the answer and in some no idea how to do! will be very grateful if you start a proper solution to Us 1) y(4)+2*a2*y''+a4*y= cos(ax) 2) y' = x + 1/y; y(0)=1 3) y' = 2*x + cos(y); y(0) = 0 4) y(x) - λ*(0 to...
  46. D

    Writing Linear Differential Equations as Matrix Differential equations

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  47. D

    3x3 Matrix Differential Equations

    Homework Statement Find the general solution to the system of differential equations. Homework Equations The Attempt at a Solution I uploaded the original equation and my work so see the attachment. I want to know how they got the vectors the got typically when I have done 2x2...
  48. F

    Second order differential equations

    Homework Statement finding the general solution. I would like to know if the 20-e^x will be treated as a sum of a polynomial? Homework Equations The Attempt at a Solution
  49. C

    Basic Differential Equations Problem (I Have No Idea What I Am Doing)

    Homework Statement Denote by L(t) the length of a fish at time t, and assume that the fish grows according to: dL/dt = k(34-L(t)) with L(0) = 2 a) solve the above equation b)Use your solution in part a to determine k under the assumption that L(4) = 10. c) Find the length of the fish...
  50. D

    Systems of Homogeneous Linear Differential Equations

    Homework Statement I uploaded the problem statement please see attachment for original problem. The problem number is 4. Homework Equations The Attempt at a Solution For clarity I uploaded what I have done please see the attachment with my work on it. I am not sure if I am doing...
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