What is Differential equation: Definition and 1000 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. W

    Bungee jump -> differential equation -> simulink simulation

    Hello guys, I found this forum using google because I need help with simulating bungee jump model. I've already done that with zero initial values and it looks good but I want it more realistic : Let's say L is the length of rope so elasticity force starts acting when y=L so logically time...
  2. F

    Solving a first order differential equation

    Homework Statement Solve the differential equation: dy/dx = 2/(x+e^y) Homework EquationsThe Attempt at a Solution I tried to use the substitution v=x+e^y, but I didn't get very far: v’=1+e^y y’ v’-1=(v-x)y' y’ = (v’-1)/(v-x) (v’-1)/(v-x) (x+v-x)=2 V (v’-1)/(v-x)=2 vv’-v=2(v-x) vv’-3v=-2x...
  3. S

    I First Order Differential Equation

    Basically, I am confused by one question in a practice paper in which the equation is given as follows: dy/dx = e^-2y and I know the general solution is equal to : y = -0.5e^-2y + C which would make sense if it was direct integration however it seems to me it is in fact separable...
  4. N

    MHB Differential equation y/(x^2+y^2)

    Can you advice the changes in this diff equation: $\d{y}{x}=\dfrac{y}{x^2+y^2}$
  5. P

    Nonlinear differential equation

    Homework Statement I need to solve the follwing differential equation $$(\frac{df}{dt}) \dfrac{4 n e^{4nf(t)}-9n e^{2nf(t)} (\frac{df}{dt})^2 + e^{2nf(t)} r^2 \frac{d^2f}{dt^2}+5n (\frac{df}{dt})^4 r^4 - r^4 \frac{d^2f}{dt^2} (\frac{df}{dt})^2}{-e^{2nf(t)}+ (\frac{df}{dt})^2 r^2}=0 $$...
  6. Mr Davis 97

    Absolute value in a differential equation

    Homework Statement ##\displaystyle (x+3)\frac{dy}{dx} = y - 2##, where x is not 3 and y is not 2. Homework EquationsThe Attempt at a Solution ##\displaystyle (x+3)\frac{dy}{dx} = y - 2## ##\displaystyle \frac{dy}{y-2} = \frac{dx}{x+3}## ##\displaystyle \int \frac{dy}{y-2} = \int...
  7. N

    MHB Solve the differential equation

    help to solve the differential equation у*у$^{\prime\prime}$=2х*((у$^{\prime})$^2)
  8. D

    Linear First-Order Differential Equation

    Hi, so I am just starting to learn differential equations and we were told that you have to put the equation into this form. In one of the examples in my lecture notes, which says "Solve (x^2-9)dy/dx+xy=0, x>3" He then says we need to rearrange this into the form given above, so he does this...
  9. L

    Non-linear elliptic differential equation - python

    Hey guys, I'm going to be honest and say I'm so stuck on this assignment - I really need help! I've took on a third year computational physics course last year - turn your weaknesses into strengths someone told me. Well, I failed and I'm back doing it again this year! So, I just have to pass...
  10. engnrshyckh

    Second order differential equation

    i am given an equation which i have to solve in simulink. the equation is quite veered to me. some one help me in understanding what kind of equation it is so that i can solve it in MATLAB simulink. (2+x^2)theta'' + (2xx' +1)theta' + 9.8(xcostheta-sinthetha)-x''=T
  11. dumbdumNotSmart

    Simple differential equation - Finding the solution

    Homework Statement We are given the following differential equation: $$ y+yy'-xy'=0 $$ Let's find the solution. Homework EquationsThe Attempt at a Solution So in my course usually we have to do some sort of substitution by the lines of x/y=z or y/x=z. This equation has proven...
  12. MidgetDwarf

    Differential Equation (Laplace transform involving a convolution)

    So I have to solve an initial value problem involving the Laplace Transformation method. I have all the terms in Y(t) besides one term, I cannot figure how to change it from frequency domain back into time domain.Not sure how to type in Latex, so i uploaded a picture, using the whiteboard...
  13. kostoglotov

    Backwards difference matrix divided by negative delta x?

    An exercise in my text requires me to (in MATLAB) generate a numeric solution to a given second order differential equation in three different ways using a forwards, centered and backwards difference matrix. I got reasonable answers for \vec{u} that agreed with each other (approximately) for the...
  14. P

    Differential Equation with Bessel Function

    <<Moderator note: Missing template due to move from other forum.>> Good afternoon. I'm trying to solve a differential equation with bessel function solutions. I am trying to solve \begin{equation*} y''(x)+e^{2x}y(x)=0 \end{equation*} using the substitution ##z=e^x##. The textbook this problem...
  15. J

    Given the solution find the differential equation

    Homework Statement For each Set write the differential equation they are a solution to. It is also asked to find the wronskian but I remember how to do that. (I am doing this on my own to brush up before I have to take mathphys next semester) 1. e-x,e-4x 2, 1,x,x2 Homework EquationsThe Attempt...
  16. J

    Determine coefficients of a differential equation

    Hi there. I've been struggling with this problem for days now (4 days, no joke) and I feel like I have a mental block and really cannot get any further. I have a system that's described by f(t) = g''(t) + 15g'(t) + 1600g(t) Where the input is g(t) The problem is to, with this information...
  17. Remixex

    Should i be worried? (DE Course)

    Hello all, my question is if i should or not be worried about the apparent "missing" (or alternative) content about my DE course. My DE course (and the only one in the list of courses i must take to get my degree) consisted on First order ODEs (Separation of variables, homogeneous, Bernoulli's...
  18. R

    Fin/Extended surface differential equation for temperature

    I'm trying to deduce the differential equation for temperature for a triangular fin: I know that for a rectangular fin, such as: I can do: Energy entering the left: q_x= -kA\frac{dT(x)}{dx} Energy leaving the right: q_{x+dx} = -kA\frac{dT(x)}{dx} - kA\frac{d² T(x)}{dx²}dx Energy lost by...
  19. Tzabcan

    What did I do wrong in solving this basic differential equation?

    I have this basic differential equation du/dt=(u^2)*(sin t) This is obviously a separable diff eq. So what I've done is: g(t) = sin t h(u) = u^2 1/(u^2) du = sin t dt Integrating both side... 1/y = - cos t + c therefor y = - 1/(cos t + c)Which is wrong, there isn't supposed...
  20. Hutchyy

    Differential equation: Autonomous equation question

    Homework Statement I'm stuck on Question #2 part A/B Homework Equations y'=r(1-y/k)y-h=y^2-ky+kh/r y''=2y-k Roots for y'= (k+/-sqrt(k^2-4kh/r))/2 I am assuming the positive root is y2 h<rk/4 [/B] The Attempt at a Solution on part a I'm getting the roots to be y2=(K+sqrt(k^2-4kh/r))/2 and...
  21. Linder88

    Ordinary differential equation with boundary value condition

    Homework Statement Consider the boundary value problem \begin{equation} u''(t)=-4u+3sin(t),u(0)=1,u(2)=2sin(4)+sin(2)+cos(4) \end{equation} Homework Equations Derive the linear system that arise when discretizating this problem using \begin{equation} u''(t)=\frac{u(t-h)-2u(t)+u(t+h)}{h^2}...
  22. H

    Power series solution, differential equation question

    I can not find a solid explanation on this anywhere, so forgive me if this has been addressed already. Given something like y''+y'-(x^2)y=1 or y''+2xy'-y=x, how do I approach solving a differential with a power series solution when the differential does not equal zero? Would I solve the left...
  23. P

    Solve a 2nd order Ordinary Differential Equation

    Homework Statement Y''-((Y')^2)+(C1*exp(Y))=C2 C1 and C2 are constants. exp = e Homework Equations No clue how to start this The Attempt at a Solution Y'=A=dY/dt Y=At+C3 (not sure) A'-(A^2)+C1exp(At+C3)-C2=0 A'-(A^2)+C1exp(C3)exp(At)=0 let C=C1*exp(C3) A'-(A^2)+Cexp(At)=0
  24. T

    MHB Solving 2nd order differential equation numerically

    I'm writing a paper about the projectile motion with the consideration og air resistance - I have obtained two formulas: ax = k*(vx2+vy2)0.5 * vx ay = k*(vx2+vy2)0.5 * vy - g (K and g are constants; K = -0,02, g =9,82) I cand write these two as 2 different differential equations: v'x(t) =...
  25. N

    Find the approximate linear ODE system

    dx/dt = x-y^2 dy/dt= x^2 -xy -2x For each critical point, find the approximate linear OD system that is valid in a small neighborhood of it. I found the critical points which are (0,0),(4,2),(4,-2) but have no idea how to do the above question! please help!
  26. Z

    Approximate solution of differential equation

    Differential equation: F(y'',y',y,x)=0, y=y(x). Now, there is g=g(x) with F(g'',g',g,x)=δ, where δ is small. Then, can g(x) be taken as an approximate solution of F(y'',y',y,x)=0?
  27. T

    Linear Differential Equation - Initial Value Problem

    Hello, I'm struggling with a simple problem here. It asks me to solve the following initial value problem: So far I've calculated the integration factor μ(x) = ex-x2 and I multiplied both sides of the equation by it and got this...
  28. A

    Can anyone make a differential equation from the below eg:

    [ NOTE ] Thread moved to homework forums by mentor suppose in a dark room a candle is burning, so darkness increases as we move away from the candle. from the below diagram can anyone derive a differential example to show the rate of change of darkness from candle to point B. supposing...
  29. T

    General Question About a System of DE's

    We were looking at some examples of systems of differential equations in class a couple days ago, and some of them were simple looking systems. One such was ##\frac{dr}{dt} = 4j## ##\frac{dj}{dt} = r## Why is it that we can't treat this as a scenario where both sides of each equation could...
  30. Remixex

    How Do I Solve a Second Order ODE with Non-Constant Coefficients?

    Homework Statement OK, this differential equation was technically created by me, because i need to clear my doubts. Y'' + sqrt(X)*Y' + X^3*Y=3sin(x) and actually just any initial conditions as long as the solution is something i can understand, let me expand my doubt further. I've never solved...
  31. M

    MHB Solutions of a linear differential equation of second order

    Hey! :o I want to check if a linear differential equation of second order has a solution in the ring $\text{Exp}(\mathbb{C})$. We define $\text{Exp}(\mathbb{C})$ as the set of expresions $$\alpha=\alpha_1 e^{\mu_i x}+\dots \alpha_N e^{\mu_N x}$$ where $\alpha_i \in \mathbb{C}$ and $\mu_i \in...
  32. M

    MHB Non homogeneous differential equation - Particular solution

    Hey! :o When we have the non-homogeneous differential equation $$ay''(x)+by'(x)+cy(x)=f(x)$$ and the non-homogeneous term $f(x)$ is of the form $e^{mx}P_n(x)$ we know that the particular solution is $$y_p=x^k(A_0+A_1x+ \dots +A_nx^n)e^{mx}$$ where $k$ is the multiplicity of the eigenvalue...
  33. M

    MHB Linear differential equation of first order in EXP(C)

    Hey! :o I want to check if there is a solution of a linear differential equation of first order in the ring of exponential sums $\text{EXP}(\mathbb{C})$. I have done the following: The general linear differential equation of first order is $$ax'(z)+bx(z)=y(z) \tag{*}$$ where $x,y \in...
  34. R

    What units will this equation have?

    Let u be density of something (heat, smell, etc.). Assume the something only diffuses; there’s no convection or ballistic transport. Let’s work in one spatial dimension (x). Then u satisfies the diffusion equation, ∂u/∂t = D ((∂^2)u/(dx^2))
  35. C

    When Will the Curse of the Medicine Man Wipe Out a Tribe?

    Homework Statement This is a interesting (morbid) problem from Simmons- Calculus with Analytic Geometry. In a certain barbourous land, two neighbouring tribes have hated one another from time immemorial. Being barbourous peoples, their powers of belief are strong, and a solemn curse pronounced...
  36. B

    Charging and discharging capacitor, differential equation

    Homework Statement A circuit consists of a voltage source, voltage ##V## , a resistor, resistance ##R##, and a capacitor, capacitance ##C##, in series. (i) Show that the charge ##Q(t)## in the capacitor satisfies the equation ##R Q' (t) + Q(t)/C = V ##. (ii) Suppose that ##R##, ##C## and...
  37. S

    Solving a first order differential equation

    Homework Statement We have the equation ## (\frac{dr}{ds})^2+(\frac{l}{r})^2=1 ## and want to solve to get ## r=\sqrt{l^2+(s-s_0)^2}## Homework EquationsThe Attempt at a Solution I have worked backwards, plugging in the solution to prove that it is correct, but the closest I have gotten to...
  38. A

    A linear differential equation problem

    Homework Statement A uniform 10-foot-long heavy rope is coiled loosely on the ground. One end of the rope is pulled vertically upward by means of a constant force of 5lb. The rope weighs 1lb/ft. Use Newton's second law to determine a differential equation for the height x(t) of the end above...
  39. B

    Differential Equation problem setup

    Homework Statement Water with a small salt content (5 lb in 1000 gal) is flowing into a very salty lake at the rate of 4 · 105 gal per hr. The salty water is flowing out at the rate of 105 gal per hr. If at some time (say t = 0) the volume of the lake is 109 gal, and its salt content is 107...
  40. B

    Differential Equation - Disappearing Term

    Hi, I am struggling to find the solution to the following equation. I can't account for the exponential term, so clearly something is going wrong... 1. Homework Statement Find the general solution to ##x' = tx + 6te^{-t^2}## where ##x(t)##. Homework EquationsThe Attempt at a Solution [/B]...
  41. M

    MHB Existence of solution of a linear differential equation

    Hey! :o I want to check if we can always find a solution of a linear differential equation of first order in the polynomial ring $F[z]$. I have done the following: The general linear differential equation of first order is $$ax'(z)+bx(z)=y(z)$$ where $x,y \in F[z]$. Or is it possible that...
  42. A

    Differential equation for grav, boyant and drag force

    Homework Statement So there is a falling object, you have to take into account the boyant force, the pull of gravity and the drag force A time dependent distance equation is what we're looking for Homework Equations Fd=CdApav2/2 Where Fd is the drag force Cd is the drag coefficient A is the...
  43. H

    MHB Please help me solve this differential equation for the initial condition (0,-1)

    Please help me solve this differential equation for the initial condition (0,-1): dx/dy = ((1+x^2)^(1/2))/(xy^3) I think I'm doing something wrong because I end up with ((x^2)(y^3))/2 = ((x^2)+y)^(1/2) + c, but when plugging in the initial condition it ends up being the square root of...
  44. D

    Solve the following differential equation (by regrouping terms)

    Homework Statement (4x^3y^3-2xy)dx+(3x^4y^2-x^2)dy=0[/B]Homework Equations (4x^3y^3-2xy)dx+(3x^4y^2-x^2)dy=0 The Attempt at a Solution i expanded it as 4x^3y^3dx-2xydx+3x^4y^2dy-x^2dy=0 next we have to take the common such that there will be m(X)(xdy+ydx)[/B] m(x) is the common onee
  45. D

    MHB Non exact differential equation

    $y(x^3e^{xy}-y) \, dx+x(xy+x^3e^{xy}) \, dy=0$ change it into exact differential and help me in solving it
  46. W

    Differential equation question (I think)

    1. Homework Statement It's to do with mirages, but I don't think the physics context is too important. It's also possible that the solution doesn't involve differential equations, and my method is completely wrong. I've been given that: ##A = \frac{n(1+ay)}{ sqrt(1+(y')^2)}## where y' is...
  47. M

    Higher Order Linear Homogenous Differential Equation

    Homework Statement Hi, basically I have a boundary value problem and just want to check that my general solution is correct. x'''' + 16x = 0 Homework EquationsThe Attempt at a Solution I'm pretty sure you make a characteristic equation which would be m4 + 16 = 0. Solving this I get m to be...
  48. Y

    Solve Differential Equation with Euler's Method

    Hi evry body i would like to have an help to resolve this exercice below the followin differential equation with its initial condition dy/dt=-lambda t y(t) t>=0 avec y(0)=y0 where lambda is damping coeficient strictly positive. -find the solution of this equation with Euler's explicite and...
  49. Remixex

    Ansatz for Hysteretic oscillation

    Homework Statement Given X'' + iG^2 X + w^2 X = 0 G^2 = h/m w^2= k/m What would be a good "educated guess" to solve that differential equation? My oscillations and waves teacher asked this on a test and since I didn't see anything depending on the speed of the object X assumed it was a really...
  50. M

    So the equation becomes u' + (1/x)u = x

    Homework Statement equation (1) y' + (ylny)/x = xy we set y(x)= eu(x) Equation (1 ) proposed becomes an equation 1st order linear differential type : equation (2) u ' + p( x) u = q ( x) a) Find the equation ( 2) using the change of variable proposed above. Homework EquationsThe Attempt at a...
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