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.

View More On Wikipedia.org
Recent contents Top users
  1. K

    Need Help with Heterogeneous Differential Equation

    Hi Everyone! I am really stuck at one differential equation: (p-y)^{2}y'(x)+(x-y)(x+y-2p)=0 where p<x<1; \ 0<y<p; \ p - parameter, 0<p<1 I have a suspicion that it does not have a solution, but is there a way to prove it mathematically in this case? I would appreciate any hints on...
  2. djh101

    Finding the differential equation (initial value problem)

    Homework problem for nonlinear dynamics. Let us write xλ(t) for the solution of the initial value problem \dot{x} = f(x) & x(0) = λ where f is continuously differentiable on the whole line and f(0) = 0. a) Find the differential equation for \frac{∂x_{λ}}{∂λ}(t) I'm a little confused...
  3. M

    Differential Equation with unit step function and Laplace transforms

    EDIT: Nevermind I see what I did wrong near the end. Homework Statement x'' + 4x = f(t) Where f(t) is 1 if t is between 0 and π, 0 if t > π. Initial conditions are x(0) = x'(0) = 0. Homework Equations Transform of a derivative: L(f^{(n)}(t)) = s^nF(s) - s^{n-1}f(0) -...-f^{n-1}(0)...
  4. D

    System of Differential equation to solve.

    Homework Statement Solve the following differential equation. y'= {{2,-1},{3,-2}}y + {{1},{-1}}(e)^{x} If it's not clear, I made an image for it. http://i.imgur.com/lypTxqf.jpg Homework Equations y{g} = y{h} + y{p} The Attempt at a Solution So basically, I am looking for...
  5. N

    Finding the equilibrium points of a differential equation

  6. P

    Solution to Bessel's Differential Equation

    Hey everyone. Need some more pairs of eyes for this one: "For each positive integer ##n##, the Bessel Function ##J_n(x)## may be defined by: J_n(x) = \frac{x^n}{1\cdot3\cdot5\cdots(2n-1)\pi}\int^1_{-1}(1 - t^2)^{n-\frac{1}{2}}\cos(xt)dt Prove that ##J_n(x)## satisfies Bessel's...
  7. N

    Solving a Differential Equation: r(y) = 8-7.7*e^(y-1)

    Homework Statement differential equation (dr/dy)+r=8 ; r(1)=0.3 The Attempt at a Solution (dr/dy)=(8-r) ∫[dr/(8-r)]=∫dy ln|8-r|+c1=y+c2 ; k=c2-c1 ln|8-r|=y+k ; r(1)=0.3 then take of e^ both sides which gives 8-r=e^(y+k) 8-e^(y+k)=r and we know r(1)=0.3 which means...
  8. N

    Seperable Differential equation

    dm/ds=m ; m(1)=7 when i find the diff eq ∫dm/m=∫ds ln|m|+c1=s+c2 ; k=c2-c1 ln|m|=s+k e ^ both sides m(s)=e^(s+k) ; m(1)=7 m(1)=e^(1)*e^(k)=7 i am stuck here. not sure how to proceed please help.
  9. D

    Circuit differential Equation

    Homework Statement The power supply in the circuit shown has V(t) = (120V)cos(ωt), where ω = 310 rad/s. Determine the current flowing through the resistor at time t = 9.7 s, given R = 600 Ω, C = 18 mF, and I(0) = 0 A. As a reminder, Kirkhoff’s voltage law for this circuit (Eq. 8-1.3 in the book)...
  10. D

    Differential equation with sphere

    Homework Statement A spherical mothball of original radius 1.2 cm slowly evaporates such that 180 days later, its radius is only 1 cm. Physically, the rate of evaporation dr/dt is proportional to the surface area of the sphere. Determine a) the time required for the radius of a new mothball to...
  11. W

    Logistic growth model, differential equation

    Homework Statement dY/dt = y(c - yb) C and B are constants. Im supposed to find and explicit solution for y, but I am having trouble. Homework Equations The Attempt at a Solution dY/y(c - yb) = dt ∫(1/c)dy/y + ∫(b/c)dY/c - yb = ∫dt (i used partial fraction decompositions)...
  12. P

    MATLAB Differential Equation formulation (and execution in matlab) problem

    Hello, I am trying to solve an equation and then execute it in matlab. It is dx/dt=I(S*A*x). so I am trying to see the evolution of x with time. The variables are all in an excel spreadsheet in different columns and the name of the columns are the name of the variables. 'I' is in an...
  13. S

    Four differential equation problems

    Hi, I need help in 4 questions. I need their answers if possible.
  14. M

    Question regarding Bernoulli Differential Equation

    In a video I was watching regarding how to solve these, the lecturer said that the form of a Bernoulli Differential Equation is y'+P(x)y=q(x)y^n where n>1 This means that if n = 1, it wouldn't be a Bernoulli differential equation and would be a first order linear differential equation, but if...
  15. M

    How does one get the solution to the differential equation for SHM?

    I understand the derivation for the simple harmonic motion equation: F = -kx ( in a 1-D case) acceleration = x''(t) = (-k/m)x so x''(t) + (k/m)x = 0 But why is the solution to this equation x = A cos (wt + ∅ ) How does one come up with this solution? I tried understanding this by reading...
  16. E

    Solution of this differential equation

    Whatsup guys, im new in this forum so be easy on me... I've been banging my head against a wall lately... i have come up across a differential equation i haven't been able to resolve... does any of you know any way of resolving this equation?? a(x) needs to be differentiable for any x belonging...
  17. A

    Spring Differential Equation w/ Damping

    Homework Statement A spring with a spring constant of 20 pounds per foot is loaded with a 10-pound weight and allowed to reach equilibrium. It is then displaced 1 foot downward and released. If the weight experiences a retarding force in pounds equal to four times the velocity at every point...
  18. Logan Land

    MHB Solving a first order differential equation

    cosx(dy/dx) + ysinx = sinx cosx (dy/dx) + y/cosx = 1 e^integral (1/cosx) ? I feel like this has to do with ln again but not sure
  19. Logan Land

    MHB Solving a linear first order differential equation

    4y'=e^(x/4) + y First I need to divide through by 4 correct? To obtain y'=(e^(x/4))/4 + (y/4) But then when I try to find integrating factor I just come up with e^(x/4) which I think is incorrect
  20. J

    Solving differential equation from variational principle

    I have the following differential equation which I obtained from Euler-Lagrange variational principle \frac{\partial}{\partial x}\left(\frac{1}{\sqrt{y}}\frac{dy}{dx}\right)=0 I also have two boundary conditions: y\left(0\right)=y_{1} and y\left(D\right)=y_{2} where D, y_{1} and y_{2} are...
  21. D

    Subspace theorem; differential equation for a subspace

    I can't seem to work out this question because it's so weird The set F of all function from R to R is a vector space given the diffential equation f"(x)+3f'(x)+x^2 f(x) = sin(x) is a subspace of F? Justify your answer I know that we have to proof that it's non-empty 0. The zero vector has to...
  22. marellasunny

    Material on how to go from data to differential equation

    Is there any material or book that explains how one could go from data to differential equation comprehensively? More like functional data analysis+differential equations
  23. F

    Find equilibrium points given 2 differential equation

    Homework Statement \dot{x} = -pxy + qx, \dot{y} = rxy - sy where p,q,r and s are positive constants (p does not equal r) Question is : Determine all the equilibrium points for the system of differential equations given above, expressing your answers in terms of p,q,r and s The...
  24. S

    RLC Circuit Second Order Differential Equation

    Homework Statement Hi there guys I am new to this forum and i have a problem with a bit of cw. It's regarding an RLC circuit. I've come up with a picture (attached) that denotes the equation. Homework Equations I know the equation is L C \frac{d^2 i}{d t^2} + \frac{L}{R} \frac{di}{dt}...
  25. A

    How Do You Solve the Toda Lattice Differential Equation for a Single Soliton?

    toda is a chain of particles of displacement q(n,t) acoplated by a spring the differential equation are \frac{d^2q(n,t)}{dt^2}=e^{-(q(n,t)-q(n-1,t))}-e^{-(q(n+1,t)-q(n,t))} the solution for one soliton is: q= Cte+ log (\frac{1+cte2 e^{-2cte3 n + 2 sinh t}}{1+cte2 e^{-2cte3 (n+1) + 2...
  26. A

    Solving a Differential Equation with a Constant and Initial Conditions

    Hello I need help with the following differential equation: (1-\frac{gh}{c^2}) A(u) - \frac{h^2}{3} A''(u) - \frac{3}{2h} A(u)^2 =0 with g,h,c=constant the answer has a \sech^2 with A(0)=A_0 and A'(0)=0 thanksution[/b]
  27. J

    Can we use y=vx for non-homogeneous differential equation?

    Homework Statement Can we use y=vx for non-homogeneous differential equation? Example: yy'=x^3+(y^2/x)→not homogeneous Homework Equations y=vx dy/dx=v+x(dv/dx) The Attempt at a Solution By substituting the equation above: vx(v+x dv/dx)=x^3+(v^2 x^2)/x v^2*x+vx^2...
  28. M

    Need help solving this differential equation

    I've separated the variables of this differential equation and end up with dx/((a-x)^(1/2)*(b-c(x-d)^3/2)). I've tried finding the integral of this with non-trig substitution methods but cannot solve it. Any help would be appreciated.
  29. G

    Differential equation d^2f/dx^2-(3-2i)f=0

    the function obeys the differential equation d^2f/dx^2-(3-2i)f=0 , and satisfy the condition f(0)=1 and f(x)----->o ,for x-----> infinity , for f=0 calculate the value of f(∏)? Can Anybody give me any hints how to go about this problem? What I know is the following; D^2f/Dy^2=(3-2i)f...
  30. H

    Minimum value of n for non-zero 4th derivative in Euler-Bernoulli beam equation

    Prove that this equation satifies the Euler-bernoulli beam equation which is given by Cany anyone help me with this. Can wolfram alpha do it? It has so many values and I'm not comfartble with doing 4th derivitives
  31. r-soy

    MHB Solving an Exact Differential Equation (#2)

  32. r-soy

    MHB Solving an Exact Differential Equation (#1)

  33. A

    Solving a fractional differential equation numerically

    Homework Statement This is a research problem that I imagine is very similar to a homework problem. I am a PhD student in biology, and I lack the mathematical background needed to make sense of the topic. I would like to find the solution to the fractional differential equation that...
  34. B

    First Order Differential Equation Help

    Homework Statement dy/dx = 3 - 6x + y - 2xy Homework Equations dy/dx + p(x)y = c p(y) dy = q(x) dx The Attempt at a Solution Just realized where my mistake was, sorry!
  35. M

    Finding the particular solution to the differential equation

    Homework Statement Please see attachment Homework Equations Please see attachment The Attempt at a Solution Please see attachment. I am supposed to compare coefficients but it doesn't seem possible. what do I do next? (also, if i have posted this in the wrong forum...I...
  36. I like Serena

    MHB Solving a Surprising Differential Equation

    I just ran into an interesting problem... which I had wrong :eek: What are the solutions to the following differential equation? $$\frac{dy}{dx} = \sqrt y \qquad \text{with }y(0)=0$$
  37. S

    Ordinary Differential Equation

    Homework Statement Homework Statement Solve \frac{dz}{dt} + 3 t e^{t+z} = 0 Homework Equations None that I can think of... The Attempt at a Solution "Rearranging" the given question, we get: \int \frac{dz}{e^z} = -3\int t e^t dt -e^{-z} = -3 \left( t e^t - e^t \right)...
  38. J

    Solving a differential equation

    I have the following differential equation \frac{\partial}{\partial t}\left(\frac{a}{X}\right)+\frac{X}{b}\frac{ \partial Y}{\partial t}+\frac{c}{X}=0 where a, b and c are constants and X is a function of t. I want to solve it for Y analytically (if possible) or numerically.
  39. J

    Can this differential equation be solved analytically or numerically?

    I have the following differential equation \begin{equation} \frac{\partial b}{\partial x}=\frac{b-c}{c^{2}}\end{equation} where b and c are both functions of x. However, although I have a closed form relation between c and x, I do not have such a closed form relation between b and x...
  40. S

    MHB Simultaneous differential equation

    Solve: dx/dt +2y+ex=t2 dy/dt-x+xex=0 Please help for this simultaneous differential equations
  41. Z

    Is this a differential equation and what do I need to be able to

    Are the equations listed below differential equations? If so, to understand them what level of calculus do I need? I only did alegbra 2 in high school. Δ Wi = η * (D-Y).Ii I(n,EF)=OF(n,EF)*I(n-1,EF) I(1,EF)=OF(1,EF)
  42. M

    Form the differential equation & homogeneous

    Hi
  43. A

    Solving a differential equation w/ undetermined coefficients

    Homework Statement y''+2y'+y=xe-x Homework Equations Yc=c1e-x+c2xe-x relevant info on textbook: "If any term of yp is a solution of the complementary equation, multiply yp by x (or by x2 if necessary)." >> i don't understand the part where it says "a solution of the complementary equation"...
  44. Fernando Revilla

    MHB Lulu M's question at Yahoo Answers (Third order linear differential equation)

    Here is the question: Here is a link to the question: Third order linear differential equation? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  45. S

    Non-homogenous differential Equation

    Homework Statement solve: y""+6y'+9y=e-3x/x3 Homework Equations y=yc+yp The Attempt at a Solution I found yc=C1e-3x+C2xe-3x and am having difficulties finding yp. I am wondering which method would be the best to determine yp: - annihilators - undetermined coefficients -...
  46. R

    Solving a 2nd Order Differential Equation with Initial Conditions

    Homework Statement \frac{d^2 y}{dx^2}\cdot\frac{dy}{dx}=x(x+1), \hspace{10pt} y(0)=1, \hspace{5pt} y'(0)=2 Homework Equations None I can think of... The Attempt at a Solution The only thing I even thought to try was turn it into the form: \frac{d^2 y}{dx^2}{dy}=x(x+1){dx}...
  47. S

    Homogenous Ordinary Differential Equation

    Homework Statement x2y"-(x2+2x)y'+(x+2)y=0 known solutions: y1(2)=2 y1'(2)=1 y2(2)=2e2 y2'(2)=3e2 Determine the wronskian Homework Equations yc=C1er1x+C2er2x I also know how to find the wronskian via a determinant The Attempt at a Solution I have tried to divide out...
  48. N

    (Differential equation) Finding an exponential equation

    Homework Statement A 44 gallon barrel of oil develops a leak at the bottom. Let A(t) be the amount of oil in the barrel at a given time t. Suppose that the amount of oil is decreasing at a rate proportional to the product of the time elapsed and the amount of oil present in the barrel. a...
  49. S

    Second Order Differential Equation (homogenous)

    Homework Statement Given y_1(x)=x is a solution to (2x-1)y''-4xy'+4y=0, find y(2) given (y(1),y'(1))=(0, 0). Utilize method of reduction of order. I need help with this as I end up getting some ugly (in my mind, anyways) integrals. Thanks in advance!The Attempt at a Solution Let y=y_1v=xv...
  50. B

    Strange Differential Equation: Solving a Radical Splitting ODE

    Homework Statement Solve the following differential equation, with the initial-value y(2)=2. y' = \sqrt{ \frac{1-y^{2} } { 1-x^{2} } } Homework Equations The Attempt at a Solution This is a strange ODE. It is continuous when either both |x|<1 and |y|<1 or when both |x|>1 and |y|>1. (Both of...
Back
Top