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. Saladsamurai

    Navier Stokes EquationQuestion about the Diff EQ

    Hello! :smile: I am going over an example in my fluid mechanics text and I am confused about a few lines. My question is more about the math then the fluid mechanics. In fact, I doubt you need to understand the FM at all; if you understand Diff eqs, you can probably answer my question. I am...
  2. D

    Solve 2nd Order Diff Eq: y''=sin(3t)+4e^t, y(0)=1, y'(1)=0

    Homework Statement find a general solution to the eq: y''=sin(3t)+4e^t Find a particular solution that satisfies y(0) = 1 y'(1) = 0 Homework Equations The Attempt at a Solution ive figured the general solution to be y(x)= -sin(3t)/9 + 4e^t + c + d And thus y'(x) = -cos(3t)/3...
  3. W

    Diff Eq - Should be easy, but my answer is upside down

    Homework Statement Find vz(r) Boundary Conditions: 1. vz(Ro) = 0 2. vz(Ri) = W Homework Equations (1/r) d/dr [r dvz/dr] = 0 (from previous problem) Let v = dvz / dr d/dr [r v] = 0 r v = c1 v = c1 / r vz = c1 ln r + c2 BC 1: 0 = c1 ln Ro + c2...
  4. K

    Solving a Diff Eq with Substitution

    Homework Statement Solve the given differential equation by using appropriate substitution. 3(1+t^{2})\frac{dy}{dx} = 2ty(y^{3} - 1) Homework Equations y = u^{\frac{1}{1-n}} The Attempt at a Solution \frac{dy}{dt} = \frac{2ty^{4} - 2ty}{3 + 3t^{2}} \frac{dy}{dt} + \frac{2ty}{3 + 3t^{2}}...
  5. T

    Diff EQ Repeated Complex Eigenvalues?

    [b]1. What dimensions of a matrix will give repeated complex Eigenvalues? Give an example of one and show that it has repeated complex Eigenvalues. [b]2. No really equations needed? The Attempt at a Solution My attempt is a 2x2 which i don't think is right but here it is. If...
  6. F

    Are there multiple valid solutions for the given initial value problem?

    (2x-y)dx+(2y-x)dy=0 y(1)=3 solve the given initial value problem and determine at least approx where the solution is valid? this should be simple right? I got 2y+2x=C and then 2y+2x=8. It doesn't match the answer in the back of the book at all, and I do see how they could have...
  7. D

    Diff Eq problem, Laplace Transform

    Homework Statement Find the Laplace Transform of the given function H(t-1)t^2 I'm not sure how to add (t-1) to the t^2 term to solve the problem Any help would be greatly appreciated.
  8. S

    Diff EQ: Mixing Problem What am I doing wrong?

    Homework Statement Alright, I know this has been done in various incarnations but I haven't found an example exactly similar to mine.. A tank contains 60kg of salt and 1000L of water. Pure water enters the tank at the rate of 8L/min. The solution is thoroughly mixed and drains at a rate...
  9. Philosophaie

    How to Estimate with a Computer program a 3rd order diff eq

    How to Estimate with a Computer program a 3rd order differential equation with 4 variables and 4 equations? The equations are derived from the geodesics equation and the Schwarzschild metric. The variables are r, theta, phi and t. Any hints would be appreciated.
  10. N

    Diff Eq vs Calc 3: Which is Harder?

    I really do apologize if this is in the wrong place. A long story short, At my school Calc 3 is a 4 creidt course with a lab and I really don't have room to fit it in my schedule next semester. But I can fit in Elementary Diff Eq. My question would be is it better to take a (much) harder...
  11. U

    Diff Eq: Variation of Parameters for 3rd-ODE's

    Homework Statement http://img27.imageshack.us/img27/6083/variationofparametersfop.jpg
  12. S

    Diff EQ- exponential raised to the exponential?

    Homework Statement Solve the equation: (x+xy^{2})dx + e^{x^{2}}y dy =0 Homework Equations n/a The Attempt at a Solution i) xdx(1+y^{2}) = -e^{x^{2}}y dy (multiply both sides by 2 to prepare for integration: ii) \frac{-2xdx}{e^{x^{2}}}= \frac{2y dy}{1+y^{2}} integrate and get: iii)...
  13. M

    Should be a simple diff eq but my answer is different than the books

    Homework Statement dy/dt = 3+e^(-t) - 1/2*y Homework Equations The Attempt at a Solution how come i got y = 6e^(1/2t) + 2e^-t - 3 when the book has y = 6-2e^(-t) - 3e^(-t/2)
  14. M

    Second Diff Eq question of the day

    Homework Statement 10. If y(t) = -te^(3t) is a solution to y'' + py' + qy = 0, then what are p and q? Homework Equations The Attempt at a Solution so it's equal to r^2 + pr +q = 0, but then how can i find the two numbers? thanks
  15. M

    Solving Separable Differential Equation: dy/dx = (6x^2)/((1+x^3)y)

    Homework Statement dy/dx = (6x^2)/((1+x^3)y) Homework Equations The Attempt at a Solution it's a separable func. so after integrating, it looks like this (base on my calculation) y^2 = 36ln(1+x^3) so how do i find y? is it just sqrt(36ln(1+x^3)) thanks
  16. R

    Non linear second order diff eq

    I'm trying to find a substitution that works for the nonlin diffeq y''+a(y')^2-by=0 Any suggestions?
  17. K

    Solving First Order Diff Eq: y^2 = x^2 + cx^3?

    Homework Statement dy/dx = (2x + y - 1)^2 Homework Equations The Attempt at a Solution Let u = 2x + y du = 2 + dy/dx dy/dx = du/dx - 2 dy/dx = (2x + y - 1)^2 so du/dx - 2 = (u-1)^2 du/dx = (u-1)^2 + 2 du / ((u-1)^2 + 2) = dx 1/sqrt(2) * arctan...
  18. 2

    How do I find eigen vectors for a repeated root case in linear systems?

    i am trying to study for a DE test and was wondering if you could help with some problems dY = -2 1 dt 0 2 i found the eigen values to be lamda = 2 i need some help on how to find the eigen vectors is this a reapeted root case and if it is how do i start off would...
  19. J

    Solving 1st Order Diff EQ: Check My Work for dy/dx = (3x+2y)/(3x+2y+2)

    Homework Statement \frac{dy}{dx} = \frac{3x+2y}{3x+2y+2} 2. The attempt at a solution I thought I'd let u = 3x+2y+2, which would give me this, u' = 5 - \frac{4}{u}. This is seperable, and we thus get \frac{1}{5}(3x + 2y +2) - \frac{4}{25}\ln|5(3x+2y+2)-4| = x + C Can someone check this...
  20. U

    General Solution for x'=ax+f(t) in Reals with Particular Solution x_p(t)

    Homework Statement x'=ax +f(t) where a & x are elements of the reals, a is a constant Let x_p be any particular solution of the equation. Show that the general solution is x(t)=x_p(t) +ce^at Homework Equations Where should I start with this? I know integrating x'=ax gives the...
  21. H

    How do I incorporate the initial velocity into the equation?

    Ok here's my problem: The acceleration of a car is proportional to the difference between 250 km/h and the velocity of the car. If this machine can accelerate from rest to 100 km/h in 10s, how long will it take for the car to accelerate from rest to 200 km/h? Here is what I've done so...
  22. J

    Y'' - y' = e^x [2nd order nonhomogenous diff Eq]

    I have an equation I need to solve by using undetermined coefficients: y'' - y' = ex The auxiliary equation is: r2- r = 0 , so 2 real roots (R1=0, R2 = 1) So, yc(x) = C1 + C2ex Now for the particular solution: I can try Aex but this is already present in the complementary...
  23. J

    V substitution in homogeneous equations (diff eq)

    Hey all, i think I'm doing most of this right, but I'm missing a coefficient somewhere when integrating or something... Homework Statement Substitute v=y/x into the following differential equation to show that it is homogeneous, and then solve the differential equation...
  24. K

    Diff EQ reduction of Order problem

    Homework Statement One solution y1(t) of the differential equation is given a) Use the method of reductions of order to obtain a second solution y2(t) b) Compute the wronskian formed by y1(t) and y2(t) Homework Equations (t + 1)^2y'' - 4(t+1)y' + 6y = 0 y1(t) = (t+1)^2 The...
  25. A

    Exploring the Possibilities of Modeling with Differential Equations

    I have read some papers where they say they are using a computational approach to modeling something complicated, like a simplified stock market or something, and they usually say that one of the reasons they are using a computational approach is becuase it would be "very difficult" to come up...
  26. A

    Proof of Integration Factor for diff eq

    Homework Statement Looking for help on this Exact first order differential equation problem. Thanks Show that if ((partial M/partial t)-(partial N/partial y))/M=Q(y) then the differential equation M(t,y)+N(t,y)dy/dt=0 has an integrating factor μ(y)=exp(integral Q(y)dy)...
  27. S

    Solving Sinusoidal Diff EQ: Lagrangian Equation Problem

    Homework Statement \ddot{\Theta}=C \sin{\Theta} where \Theta is a function of time, and C is a constant. I ran into this on a Lagrangian Equation problem, and though the problem doesn't ask for the solution, I'm wondering how one would solve this Diff EQ. I'm afraid my intro to Diff EQ...
  28. B

    Solving H.S. Bear's Diff Eq Problem: (1-y^{2}) dx - xy dy = 0

    I am reviewing differential equations, going through H.S. Bear's book Diff Eq: Concise Course. The problem set for the variables separate section were pretty easy and straightforward except for this one, which I can't see how to arrive at the answer given in the book. I'm probably just missing...
  29. H

    Mastering Critical Damped Systems: Diff EQ Spring Question [SOLVED]

    [SOLVED] Diff EQ spring question Homework Statement a)IF a mass of 0.5kg is attached to a spring with a spring constant of 5(nt/m) and then receives a blow to dislodge it from its equilibrium position, then what is the resistive force coefficient (gamma) if the system is critically damped...
  30. H

    Diff Eq Problem y'' - 3y' - 4y= 5e^-x - 3x^2 + 7

    [SOLVED] Diff Eq problem Homework Statement y'' - 3y' - 4y= 5e^-x - 3x^2 + 7 Homework Equations I think I would need to find complimentary solution, then the particular solution using variation of parameters y=y(c) + y(p) The Attempt at a Solution y(c)= r^2-3r-4=0...
  31. H

    Solving Tough Differential Equations with Variation of Parameters | [SOLVED]

    [SOLVED] Tough Diff EQ problem Homework Statement y^(4)-8y^(2)-9y=7e^(-3x) y^(4)=fourth derivative, y^(2)=second deriv. Homework Equations Need to find complimentary solution and particular solution using variation of parameters The Attempt at a Solution r^4-8r^2-9=0...
  32. B

    Conversion of System of Eq's to 2nd Order Diff Eq

    My question is in regards to converting a system of differential equations into a higher order differential equation. I am an undergrad taking diff eq and have just learned the wonders of Euler's method of solving 2nd order differential equations with constant coefficients. It is significantly...
  33. S

    Diff Eq. - swapped coefficients help

    Homework Statement In an L-R-C circuit, L = 1, R = 2, C = 0.25, E(t) = 50 cos(t) Find the steady state solution Homework Equations The Attempt at a Solution L \frac {di(t)}{dt} + R \frac {dq(t)}{dt} + \frac {q}{C} = 50cos(t) \frac {di(t)}{dt} + 2 \frac {dq(t)}{dt} + 4q...
  34. H

    Diff EQ Application: Alcohol Concentration in a Tank

    [SOLVED] Diff EQ applications problem A tank initially holds 25gallons of water. Alcohol enters at the rate of 2gallons/minute and the mixture leaves at a rate of 1 gallon/min. What will the concentration of alcohol be when there is 60gallons of fluid in the tank The Attempt at a...
  35. F

    Linear (I think?) First Order Diff EQ

    So this equation came up: xy' + y = cos x Now I was just wondering how to solve this, all I've learned how to do is separation of variables, which cannot be used in this case. Basically I ask this because a solution for y is an infinite series, so basically I'm just wondering if the...
  36. Saladsamurai

    Power Series Solution to a Diff EQ

    [SOLVED] !Power Series Solution to a Diff EQ! Homework Statement Find the first 5 term of a Power series solution of y'+2xy=0 (1) Missed this class, so please bear with my attempt here.The Attempt at a Solution Assuming that y takes the form y=\sum_{n=0}^{\infty}c_nx^n...
  37. C

    2nd order diff Eq help

    2nd order diff Eq with t missing I am trying to find y as a function of t and y'' - y = 0 The two IV given are y(0) = 7, and y(1) = 5 .. Remark: the initial condition involves values at two points. Well since y = {y,y''} and the independent variable t does not appear, I went about it by...
  38. T

    Two Questions (antiderivative and diff eq.)

    Homework Statement (i've put my questions in bold) The differential equation is dy/dx = xy^2 I separated variables and came up with (solving for c) c = -1/y - (x^2)/2 I then had to find the particular solution for y(0) = 1, so I came up with c = -1 - 0 so c = -1 I then...
  39. J

    Reducing Order of Differential Equation: t^2y'' - 4ty' + 6y = 0 [SOLVED]

    [SOLVED] Reduce Order (diff eq) nevermind i got it, cliffsnotes ftw Homework Statement Solve the differential equation using the reduction of order method. t^2 y'' - 4ty' + 6y = 0 t > 0 y_1 (t) = t^2 Homework Equations The Attempt at a Solution Well The first thing I do is...
  40. Saladsamurai

    Can You Solve This Third Order Homogeneous Differential Equation?

    Is this diff eq solvable? ! Homework Statement Find the general solution to the third order homogeneous diff eq if one solution is known to be: x^2e^{5x} I was thinking of using reduction of order but I don't have the original equation!
  41. D

    Forced Mass-Spring System - Diff Eq

    Homework Statement You have a forced, mass-spring system, without damping. Spring constant = 4 N/m weight of mass = 9.8 N mass = 1 Kg Find the motion X(t) of the mass if ω = 1.5 (Hz) and deduce the maximum elongation of the spring. Sketch the vibrations X(t). Do same for ω = 1.9, 3. Find...
  42. Saladsamurai

    Question about the General form to normal form of Diff Eq

    Homework Statement The diff eq [itex]x(y')^2-4y'-12x^3=0[/tex] takes the general form. Determine if the equation can be put in the normal form dy/dx=f(x,y) Well I have tried algebraically to isolate y' and find that I cannot. So my question is, I have decided the answer is no. Now how do...
  43. KleZMeR

    Second Order Linear Diff Eq

    Quadratic Polynomial can't satisfy second order linear differential equation with constant coefficients... WHY? and if possible, how would I show this? Does it have to do with independence?
  44. P

    Master Diff Eq: A Beginner's Guide to Understanding Intermediate Steps"

    Hello guys, I want to learn diff eq but I seem to not be seeing many of the intermediate steps. Is there a guide or tutorial that gives one a good intro to diff eq with all the intermediate steps? Thanks
  45. P

    Mastering Differential Equations: A Comprehensive Guide to Intermediate Steps

    Hello guys, I want to learn diff eq but I seem to not be seeing many of the intermediate steps. Is there a guide or tutorial that gives one a good intro to diff eq with all the intermediate steps? Thanks
  46. A

    Solving Diff Eq Tired Spring Problem: Find Min k for y(x) Cross y=0

    Homework Statement Determine the minimum positive value of the constant k needed to ensure that the function y(x) obeying x^2 y'' + K y = 0, y(1) = 0, y'(1)=1 will not simply keep on increasing toward large x but will instead return to cross y=o ocassionally The Attempt at a...
  47. A

    Diff Eq prob from advanced dynamics

    Homework Statement Diff eq z*z(double dot) + z(dot)^2 - gz = 0. The dots are derivatives with respect to time. Initial condition z(0)=z(not) Homework Equations The Attempt at a Solution This is part of a larger advanced dynamics problem, where I am asked to get the velocity (z...
  48. B

    Simple Differential Equation Help | Solving y'[x] = xy | e^x Solution Explained

    Homework Statement y ' [x] = xy y[0] = 1 Homework Equations none The Attempt at a Solution I posted this somewhere else and 2 people got the answer is y = e^x I am getting something else though. I get up to here: dy/dx = xy 1/y dy = x dx ln y = (x^2)/2 + C...
  49. N

    Struggling with Solving a Diff Eq? Get Expert Help Here!

    Never expected to be pleading for help so soon, and especially not on a differential equation, which I usually am good at. But for whatever reason, I cannot solve this problem: y*d(y,x,2) + (d(y,x))^2 + 1 = 0 Any help would be greatly appreciated! ETA: I know I'm supposed to substitute...
  50. J

    VI Arnol'd Diff Eq Question about first problem.

    Sorry guys, I am not too bright and I am trying to learn diff eq on my own (have the class next year) and I am hung up on this problem: If the density of air at sea level is 1250g per cubic meter, at what altitude is the density half of that? The answer is 8ln(2) but I don't know why. I'd...
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