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. F

    Diff Eq Undetermined Coefficients

    Homework Statement Find the solution for y"+2y'+5y=(e^x)sinx Homework Equations The Attempt at a Solution So far I think I've gotten the solution from the characteristic equation, but I'm having trouble with the particular solution. For the characteristic equation solution...
  2. G01

    Solving Diff EQ using a Laplace Transform

    Homework Statement Solve the initial value problem: \frac{dy}{dt} + 2y = u_2(t)e^{-t} y(0) = 3 Where u_2(t) is a Heaviside Function with the discontinuity at t=2. Homework Equations The Laplace transform of a Heaviside function multiplied by another function: L( u_a(t)f((t-a) ) =...
  3. R

    How Do You Solve the Differential Equation kxy = (y+dx)(x+dy)?

    I have run into this problem solving differential equations of this type (they occur often doing momentum problems): kxy = (y+dx)(x+dy) where k is constant. I multiply it out to : kxy= xy + xdx + ydy + dydx Regroup and : [tex] \int {kxy} = \int {xdx} + \int {ydy} + \int {dydx} [/itex]...
  4. M

    Finding ZIR given a diff eq

    Homework Statement Calculate ZIR(t) for the system described for the following differential equation, where the initial condition is y(0-) = 5: Homework Equations 2y' + 3y = 2x' + x(t-1) The Attempt at a Solution I'm pretty sure we're supposed to laplace-transform it, then find Y/X...
  5. T

    Very simple diff eq [concept help needed for exam]

    Homework Statement Find the exact solution to this problem y'=4x–y+9;y(0)=6 The Attempt at a Solution I am panicing because I have an exam tomorrow and I can't remeber a lot of the basics for diff eq. I tried to solve this using 2nd order style... first I made it y'-y=4x+9...
  6. G

    Second order diff eq with two variables

    Here's the problem: x^2y''-3xy'-12y=0 with initial conditions y(1)=0 and y'(1)=7 I'm supposed to solve for y in the form y=c1y1+c2y2 y1 = x^6 by inspection Now to solve for y2 y2=y1v v can be solved for by the equation y1v''+(2y1'+py1)v'=0 where p is the function in front...
  7. G

    Second order diff eq with i

    Crap, nevermind, I left b^2 out of the quadratic formula, thanks anyways. Here's the question: Find y as a function of t from the diff eq: y''+6y'+25y=0 with the initial conditions y(0)=8 and y'(0)=8 I used the form r^2+6r+25=0 to solve for r and through the quadratic equation got r =...
  8. G

    How Do You Solve a Homogeneous Differential Equation with Repeated Roots?

    Never mind, I figured it out. Here's the question: Find the general solution to the homogeneous differential equation https://webwork.math.uga.edu/webwork2_files/tmp/equations/59/540a7a16e5c4e841a098d9d2a72f0a1.png The solution has the form...
  9. D

    Transfer Function of Coupled Diff Eq

    I have two coupled harmonic oscillators: \ddot{x}_{1} = -2kx_{1} + kx_{2} + f(t) \ddot{x}_{2} = kx_{1} - kx_{2} Mass 1 is at position x_{1} and subject to force f(t). I take the Laplacian of the first equation and solve for X_{1} to get X_{1} = \frac{ F(p) + k X_{2} }{ p^{2} + 2k }...
  10. Kudaros

    Why Does This Differential Equation Epidemic Model Stump Me?

    I have a logistic issue with differential equations. I have spent four hours working on this problem and it is way past the point of ridiculous. If anyone can help out that would be great. It begins like this. dy/dt = k(P-y)*y. It is an epidemic model, where k is a positive constant...
  11. D

    How to Solve a Second Order Differential Equation with Given Initial Conditions?

    Ok, so i tried to solve this problem: Find y as a function of t if: 100Y"-729y=0; y(0)=6, y'(0)=1 this is what i did so far: 100r^2-729r=0 r(100r-729)=0 r=0, r=729/100 y(x)=C1+C2*e^((729/100)*t) y'(x)=C1+729/100C2*e^((729/100)t) am I on the correct track? After I substitute the...
  12. H

    This is from a Diff EQ but but its just algebra i think?

    this is from a Diff EQ but but its just algebra i think?? I'm just working through an example here and its say to solve the equation (y/k)=(ynot/k)/((ynot/k)-[1(ynot/k]e^(-rt)) for t. then in the next line it shows it solved for e^(-rt)={(ynot/k)(1-(y/k)}/{(y/k)(1-(ynot/k)} now i have...
  13. B

    Solve Diff Eq Problem (s+1)/s*(s^2+s+1): Get Expert Help

    (s+1)/s*(s^2+s+1) find laplace transform.. i am confused how to solve this problem i did numerator separation and i did A/s + (Bs+c)/(s^2+s+1) but still doenst work..so experts..i do really need ur help..thnx in advance
  14. tandoorichicken

    How Can y(x) Be Expressed as a Convolution in This Differential Equation?

    Any hints to this problem? "Assume the solution to a differential equation is given by \frac{dy(x)}{dx}+ay(x) = f(x) where y(0)=y_0 and a is a constant. Show how y(x) can be written as a convolution of f(x) and an exponential e^{ax}." The only hint we got from the prof was to multiply...
  15. V

    Why Is My Solution Different from the Textbook's for This Linear ODE?

    Hey, So I solved this one linear DE and the answer I got isn't the same as the one in the back of my textbook... And I'm not sure why. I thought I was doing this right. Could someone tell me what I'm doing wrong? Here's what I'm given: x^{2}y'+xy=1...
  16. B

    Question about simple diff eq question

    hi guys..i got a problem while doing my diff eq assignment:confused: hope u guys can give me hand..any help ll be appriciated Obtain the solution of the diff eq.. x' + ax = A sinωt x(0)= b thnx guys..
  17. W

    Diff EQ with the boundary conditions

    I took an ODE course last year, but I seem to have forgotten some stuff. I need to solve this equation: \frac{d^2u}{dt^2} + {\omega}^2u = f_osin({\mu}t) with the boundry conditions: u(0) = 0, du/dt(0) = 0 When I tried to solve the homogenenous equation first, I got...
  18. P

    How do I determine behavior of y at infinity for a given differential equation?

    Draw a direction field for the given differential equation. Based on the the direction field, determine the behavior of y at t goes to infinity. If this behavior depends on the initial value of y at t=0, describe the dependency. y'= -1-2y In class we did examples where we had a range...
  19. H

    Solving Diff Eq for Free-fall w/o Drag in Tank Outlet

    Suppose a tank containing a liquid has an outlet near the bottom. The velocity of the liquid flowing out of the outlet is equal to a particle free falling without drag. a. show that v=sqrt(2gh) b. by finding the rate of outflow to the rate of change of the liquid in the tank, find...
  20. L

    Diff eq of mechanical system w/ friction with zero input

    diff eq of mechanical system w/ friction with zero input... I was wondering how we get around modeling mechanical systems with frictional forces (forces proportional to normal force and in the direction opposite of the motion) and the external force is zero. So, take for example a second order...
  21. D

    Diff Eq Region of Unique Solution

    I have just started my first diff eq class, and I am struggling with the concepts. In the section 'Preliminary Theory' we are given Picard's Theorem and a few odd examples. Then this is the question from the homework: "Determine a region of the xy-plane for which the given differential...
  22. D

    Non-homogeneous 2nd order diff eq involves power series

    I just need a hint or something to see where I start. I'm at a loss for a beginning. Consider the non-homogenous equation y'' + xy' + y = x^2 +2x +1 Find the power series solution about x=0 of the equation and express your answer in the form: y=a_0 y_1 + a_1 y_2 + y_p where a_0 and...
  23. W

    Help with Diff EQ Integrals Using Trig Substitution

    I am working on a Differential Equation problem and I am stuck on these two integrals: http://forums.cramster.com/Answer-Board/Image/cramster-equation-20064101738436328028752372062504976.gif and...
  24. S

    Real-World Applications of Differential Equations

    I'm looking for any resources you guys can point me to on "real" applications for differential equations. I have a relatively decent grasp of the math, but it's very frustrating for me learning sterile math with no idea what it's used for, especially since I figure that I'm going to actually...
  25. M

    Confused on a first order seperable diff EQ wee

    Hello everyone. I'm going back to all my old webworks and trying to finish them and I'm still having problems on first order. It says this is seperable but I'm not seeing it. Here are the directions: The differential equation...
  26. M

    Stuck on power series, need a refreshment Diff EQ converge?

    Hello everyone! I remember doing these in calc II, but forgot 99% of it. Here is the question: Find the interval of convergence for the given power series. http://cwcsrv11.cwc.psu.edu/webwork2_files/tmp/equations/5e/119a4cd07ce5d3f6f673477ed169641.png The series is convergent from x = ...
  27. M

    Confused what method to use on this Diff EQ 2nd order

    ello ello! I havn't seen an example of this type of problem. It has another variabler in the equation, i tried to divide by it, but its still in there. Here is the problem: Find y as a function of x if...
  28. M

    Y''-6y'+5y = -25t^3, Diff EQ, did I mess up my guess again?

    Hello everyone! its me! I thought i for sure did this one right, but no, lies! Directions are to find a particular solution Here is what i did: http://suprfile.com/src/1/1t3y9m/lastscan.jpg Thanks :)
  29. M

    Y''+9y = 6sin(3t); Diff EQ, weee where did i go wrong?

    Ello ello! I'm looking at an example problem my professor did which is very similar to the one that i am doing, the only difference is his problem is: y''+y = sin(t) mine is: y''+9y = 6sin(3t); THe problem says: Find a particular solution to y'' + 9 y = 6sin(3 t) . y_{p} = so here is what i...
  30. M

    Variation of parameters problem, very BIG, but i think i'm right so far Diff EQ

    OKay everyone, this is a big f'ing problem (to me anyways) and its only worth 1 point! But I'm doing it anyways. So here was my attempt, everything seems to be working out like it should but look at what u1 came out too, what am i going to do with that mess? Also do u see any mistakes...
  31. M

    2nd order Diff EQ with x^2*y'', what shall i do i missed it

    Hello everyone! i'm confused on how to approach this problem, my professor did an example and he used m^2-m-4m+6 = 0, if u have t^2*y''-4ty' +6y = 0; So i tried to do the following, but the answer is wrong. Anyone see? http://img88.imageshack.us/img88/3603/lastscan7jj.jpg THanks!
  32. M

    Varation of Parameters fun diff EQ question, where do i go next?

    Varation of Parameters fun! diff EQ question, where do i go next? This is my first attempt at doing Variation of parameters, didn't go to bad, things cancled out pretty well but now I'm almost done but I'm stuck! The problem says: Find the solution of y''+15y'+56y = 54*e^(-5t), with y(0) = 8...
  33. M

    What happens if you only get 1 solution to a 2nd Order Diff EQ?

    Hello everyone, I'm slightly confused on this problem, when i factored it and solved for r, i came out with only 1 answer, r = -13/72 Here is my problem and work: http://img213.imageshack.us/img213/685/lastscan15uk.jpg :biggrin:
  34. A

    How Do You Calculate the Drag Coefficient to Decelerate a Dragster?

    intro to DIFF EQ (help!) a drag chute must be designed to reduce the speed of a 3000-lb dragster from 220 mph to 50 mph in 4 sec. Assume that the drag force is proportional to the velocity. a) what value of the drag coefficient k is needed to accomplish this? b) how far will the dragster...
  35. M

    2nd Order Diff EQ with 2 intial conditions, got complex roots, i f'ed it up

    OKay i havn't gotten 1 2nd order Diff EQ right yet! I'm on a role! wee! Find y as a function of t if 81y'' + 126y' + 79y = 0, y(0) = 2, y'(0) = 9 . Here is my work: http://img204.imageshack.us/img204/4605/lastscan5ag.jpg I submitted this and it was wrong...
  36. M

    Confused on why i'm missing this 2nd Order Diff EQ with complex roots

    Hello everyone. I"m not getting this problem right. <insert sad face here> Find y as a function of t if 6y'' + 33y = 0, y(0) = 8, y'(0) = 5 . y(t) = hokay, here is my work, it is sloppy sorry. Can you see any obvious mistakes I made? Note: the sqraure root should be encompassing both the 11...
  37. M

    Simple homogenous 2nd order Diff EQ, but i'm stuck on the process

    Hello everyone, we just started 2nd order differentials, and i was loooking at his example and it made senes but now I'm doing the homework and I'm stuck. Here is the problem: Find y as a function of t if y'' - 3y' = 0, y(0) = 9, y(1) = 7 . y(t) =? Well there is my work...
  38. M

    Diff EQ, i'm confused on plotting this solution (series)

    Hello everyone. This problem has 4 parts. I got part (a) which is suppose to be the hardest part but I'm confused on how to (b) Plot Qn(t) for n = 1...4 observe wether the iterates seem to be converging. Well this is the series solution I got for the differential equation...
  39. M

    Confused on what u do next with an exact equation, Diff EQ

    Hello everyone. We just started working with exact equations and I'm confused on what I do next! here is the problem: Use the "mixed partials" check to see if the following differential equation is exact. If it is exact find a function F(x,y) whose level curves are solutions to the...
  40. M

    Intial Value problem, Diff EQ, my steps look right but answer = wrong wee

    Hello everyone I'm stuck on this problem: http://cwcsrv11.cwc.psu.edu/webwork2_files/tmp/equations/18/0c06a4e72ea888ea65443a46afd42f1.png http://cwcsrv11.cwc.psu.edu/webwork2_files/tmp/equations/90/dff750d2ab4fefb41eccca8ce709a01.png Looks simple enough but the answer they said is wrong. Here...
  41. M

    Solving Diff EQs: Find Solution & Related Functions

    more Diff EQ fun! http://cwcsrv11.cwc.psu.edu/webwork2_files/tmp/equations/58/6217226076d5fd259f53ad1e3ed4071.png Find such a solution and then give the related functions requested. http://cwcsrv11.cwc.psu.edu/webwork2_files/tmp/equations/57/f15a6cea760ce4906a6ec16f9d29451.png I submitted...
  42. M

    Diff EQ, did i mess up on a natural log rule?wee

    Hello everyone the problem I am messing up is: . Solve the following initial value problem: http://img59.imageshack.us/img59/9037/lastscan4vl.jpg When pluggin in answers and using my calculator to evaluate, its saying NON-REAL numbers, I'm messing up hardcore somewhere but not sure where...
  43. M

    Diff EQ, can't see what I did wrong expressing this seperable

    Hello everyone. I don't see how this problem is wrong, i never do. Here it is: Solve the differential equation (y^{15} x ){dy}/{dx} = 1 + x. Use the initial condition y(1) = 3. Express y^{16} in terms of x. y^{16} = ? y^15*dy = (1+x)/x dx; I integrated both sides and got: (y^16)/16 =...
  44. M

    Diff EQ w/ I.C. anyone see where i f'ed it up?

    Its me again! I'm just having a full day of Diff EQ! 90% of the time I'm wrong but I'm learning slowly! but here is another problem: The directions say: Find the particular solution of the differential equation \frac{dy}{dx} + 5 y = 7 satisfying the initial condition y(0)=0. Answer: y = ? I...
  45. M

    Simple seperable Diff EQ w/ I.C., did i make an algebra error?

    I'm lost on this one, i checked my problem over a few times and i don't see any algebraic errors, did i break some rule? Thanks! :smile: Find the solution of the differential equation (ln(y))^4*{dy}/{dx} = x^4 y which satisfies the initial condition y(1)=e^2. y= ? Here is my work...
  46. M

    Given the y-intercept of this curve means what? Diff EQ

    Hello everyone this may be an easy problem, but I'm confused on what they have given me the problem says: Find f(x) if y = f(x) satisfies {dy}/{dx} = 32 yx^{15} and the y-intercept of the curve y = f(x) is 2. f(x) = ? But what does that mean that the y intercept is 2? is that an...
  47. M

    Problem solving separable diff EQ w/ I.Condition. , they say its wrong :x

    Hello everyone i did this one surley thing it would work out and yet another failure. :cry: Solve the separable differential equation 11 x - 8 y*sqrt{x^2 + 1}*{dy}/{dx} = 0. Subject to the initial condition: y(0) = 6. y = ?I'm pretty sure where I messed up is when i tried to solve for y, i...
  48. M

    Can't seem to figure out how to solve for C Diff EQ question

    Hello everyone I'm stuck, i got right to the end of this intial value problem and it doens't seem to want to work. Solve the separable differential equation for u {du}/{dt} = e^{3 u + 10 t}. Use the following initial condition: u(0) = 6. u = ? I might have screwed up here: du =...
  49. M

    Diff Eq problem, finding the small value and large value of k?

    Hello everyone I'm lost on how they want me to approach this: Find the two values of k for which y(x) = e^{kx} is a solution of the differential equation y'' - 16 y' + 60 y = 0. smaller value =? larger value = ? I did the following: y(x) = e^(kx); y' = ke^(kx); y'' = k^2e^(kx)...
  50. J

    Velocity and resistance problem. diff Eq type

    A body moving with velocity V enounters a resistance in the form of dV/dT=-k*V^(3/2). Show that V(t)=(4v.)/(k*t*sqrt(v.)+2)^2 v. is the same as v(subscript 0) When i solve the differential equation i keep getting V(t)=4/(k*t-v')^2 I have no idea how to get from dv/dt to the given v(t)
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