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. V

    MHB Second order inhomogeneous differential equation

    Hi MHB. I'm having yet another doubt regarding differential equations. Can someone please help me out? Thanks. Consider the following differential equation: {y}''+{y}'= x^{2} I have found the homogeneous solution to be: y_{H}=c_{1} + c_{2}e^{-x} But when finding the particular solution...
  2. W

    Solving real life problem with differential equation

    Homework Statement A car starts from rest.When it is at a distance s from its starting point,its speed is v and it acceleration is a = (25v + v^3). Show that dv = (25 + v^2)ds and find its speed when s = 0.01 2. The attempt at a solution a = v(dv/ds) = (25v + v^3) divide both sides by v...
  3. B

    Differential Equation arising from Anharmonic Oscillator

    Homework Statement Okay, I am trying to solve this Anharmonic Oscillator equation. Now I am given with the potential U=(1/2)x^2-(1/4)x^4 and Kinetic energy T=(1/2)x' ^2 So the Lagrangian becomes \mathcal L=T-U Now I have taken all the k's and m to be 1 Homework Equations...
  4. V

    MHB Differential equation for draining a pool

    Hi MHB. Can someone help me with this one please? I don't understand what the question is really saying...in particular part (c). I tried to set up dD/dt = k - D^1/2 but it doesn't seem correct. Thanks.
  5. V

    MHB Equilibrium solution limit to differential equation

    Can someone please help me with this one? I have found the equilibrium solutions,but I'm not sure what to do next. Consider dx/dt = x^3 - 4x Given a solution x(t) which satisfies the condition x(0) = 1, determine the limit(t -> infinity) of x(t). Thanks!
  6. W

    Second Order differential equation involving chain rule

    Homework Statement Solve d^2x/dt^2 = (3x^3)/2 when dx/dt = -8 and x = 4 when t = 0 2. The attempt at a solution v = dx/dt dv/dx = d^2/dx^2 d^2x/dt^2 = v(dv/dx) = (3x^3)/2 v dv = (3x^3)/2 dx integrating and using limits and you get : v^2/2 -32 = (3x^4)/8 - 96 ...
  7. S

    MHB Numerical solution of partial differential equation

    I need to solve the following system of equations for n=0,1,2 subject to the given initial and boundary conditions. Is it possible to solve the system numerically. If yes, please give me some idea which scheme I should use for better accuracy and how should I proceed. The coupled boundary...
  8. Y

    Question in Partial differential equation.

    For polar coordinates, ##u(x,y)=u(r,\theta)##. Using Chain Rule: \frac{\partial u}{\partial x}=\frac{\partial u}{\partial r}\frac{\partial r}{\partial x}+\frac{\partial u}{\partial \theta}\frac{\partial \theta}{\partial x} The book gave \frac{\partial ^2 u}{\partial x^2}=\frac{\partial...
  9. S

    Partial differential equation problem

    Homework Statement using the method of separation of variables, solve ∂u/∂x=4∂u/∂y, where u=3e^-y - e^-5y when x=0. Homework Equations The Attempt at a Solution let u(x,y)=X(x)Y(y) =XY.
  10. W

    Second Order differential equation involving chain rule

    Homework Statement Solve (d^2x)/(dt^2) = 2x(9 + x^2) given that dx/dt = 9 when x = 0 and x = 3 when t = 0 Homework Equations The Attempt at a Solution v = dx/dt ...... dv/dx = d^2x/dt^2 dv/dx = v(dv/dx) v(dv/dx) = 18x +2x^3 integrating and evaluating using...
  11. H

    Differential equation with repeated roots

    Hi, I'm somewhat new here, only posted a few times, and would like some help from you guys here if possible I'm stuck with a problem on the topic mentioned. x'=Ax A is a 2*2 matrix A = [-5 1] [-1 -3] Now I managed to find the eigenvalues which is -4, repeated twice (multiplicity 2) And the...
  12. F

    Fourth order differential equation

    Homework Statement I have this physics mathematical problem : (see link in comment) EI(∂4u)/(∂x4) =f ......(1) The boundary conditions are: ∂2u/∂x2 =0 and EI ∂3u/∂x3 =±F where E is Young’s modulus, I is second moment of area, f is force per unit length...
  13. C

    Solution of an Ordinary Differential Equation

    Hi, The definition (see attachment) says that f(x) is a solution to the differential equation if it satisfies the equation for every x in the interval. Assuming that I have a differential equation that I want to solve and the D.E. has an interval I_1, and I've come up a solution with...
  14. U

    Differential equation, linear sys. of eq. problem

    Homework Statement Right so I had a standard systems of diff eqs involving repeated eigenvalues. When I find the last vector I have the equation \begin{pmatrix} 1 & 1\\ -1 & -1 \end{pmatrix} \begin{pmatrix} u_1\\ u_1 \end{pmatrix} = \begin{pmatrix} 1\\ -1 \end{pmatrix}The Attempt at a...
  15. F

    MHB Getting wrong answer to differential equation (first order separable ODE)

    \sqrt{1-y^2}dx - \sqrt{1-x^2}dy=0, y(0)=\frac{\sqrt{3}}{2} rewriting the equation gives \frac{1}{\sqrt{1-x^2}}dx = \frac{1}{\sqrt{1-y^2}}dy Isn't this the integral for \sin^{-1}(x) & \sin^{-1}(y)? The back of book has y=1/2x+\frac{\sqrt{3}}{2}\sqrt{1-x^2}
  16. O

    Differential Equation, nonhomogeneous equation

    Homework Statement Find the general solution: y'' - y' - 2y = -2t + 4t^2 Homework Equations The Attempt at a Solution r_1 = 2, r_2 = (-1) Set Y(t) = At^2 + B^t + C Y' = 2At + B Y'' = 2A 2A - 2At + B - 2At^2 + Bt + C = -2t + 4t^2 -2At^2 + (B-2A)t + 2A + B + C = -2t...
  17. F

    MHB How to know if there's an explicit solution for differential equation

    Solve the given differential equation by separation of variables y\ln{x}\frac{dx}{dy}=(\frac{y+1}{x})^2 I got it down to \ln{x}x^3-\frac{x^3}{3}=y^3+3y+y^2 At this point I had no idea how to solve having y^3 y^2 and y terms so I did what any good student would do and checked the back of the...
  18. Petrus

    MHB Differential equation with eigenvector (complex number)

    Hello MHB, Solve the following system of linear differential equation f'=f-g g'=f+g with bounded limit f(0)=0, g(0)=1 could anyone check if My answer is correct? Just to make sure I understand correctly! ps we get \lambda=1-i and \lambda=1+i Regards, |\pi\rangle
  19. Petrus

    MHB Differential equation with eigenvector

    Hello MHB, solve this system of linear differential equation f'=f-g-h g'=-f+g-h h'=-f+g+h with boundary conditions f(0)=1, g(0)=2 and h(0)=0 we get that \lambda=1 and \lambda=0 now for eigenvector or we can call it basis for eigenvector \lambda=0 i get Is that correct? Regards, |\pi\rangle
  20. C

    MHB First Order Nonlinear Ordinary Differential Equation

    Hello, can you guys help me please with this differential equation from Demidovitch book, is to find the solution transforming to polar coordinates :
  21. D

    Differential Equation Question

    I need to prove that the solution of this differential equation: dx/dt = -x3 + 2*x + sin3(2*pi*t) - 2*sin(2*pi*t) + 2*pi*sin(2*pi*t) has the solution: ψ(t,0,0) = sin(2*pi*t) I know that I need to get all of the x's on one side and the t's on the other then integrate, but I can't...
  22. K

    Find fourth order differential equation

    Homework Statement Suppose that a fourth order differential equation has a solution y=-9e^(3x)xcos(x). a. Find such differential equation, assuming that it is homogeneous and has constant coefficients. b. Find the general solution to this differential equation. x is the independent...
  23. D

    Differential Equation for a Wien Bridge Oscillator

    I am trying to write out a differential equation for the Wien bridge oscillator circuit. I have attached a picture of the circuit. I am considering ideal conditions. I am trying to solve for the output voltage but I need help setting up the differential equation.
  24. V

    2nd Order Differential Equation via Power Series

    Homework Statement Problem 5 on the attached Sheet here Homework Equations We studied the Power Series Method and how to calculate a linearly independent solution if one solution is already known. So we need to find one solution (probably) using the power series method and then using...
  25. ThomasMagnus

    Angular Velocity Differential Equation

    Problem: Newton's 2nd Law for rotational motion states that the product of the moment of Inertia, I, and the angular acceleration alpha(t) is equal to the net Torque acting on a rotating body. Consider a wheel that is being turned by a motor that exerts a constant torque T. Friction provides...
  26. W

    First Order Differential Equation

    Homework Statement Find the general solution to the following differential equation. dy/dx = 2x( (y^2) + 1) Homework Equations The Attempt at a Solution I got all x terms on one side and all y terms on the otherside 2x dx = 1/( (y^2) + 1`)dy integrate x^2 + c =...
  27. P

    Solve a differential equation sytem. Eulers method

    Homework Statement dx/dt=x+y dy/dt=-y+8x Solve this system useing Eulers method or by elimination method. Homework Equations Eulers method I know that in general I should look for solution in form y=const*exp[kx] z=const2*exp[kx] But in my case I am not sure if i can write...
  28. mesa

    Solve a differential equation for the initial conditions,

    Homework Statement Solve y' = (y^3)(t^2) for the initial condition y(0)=0 and state in which interval in 't' this solution exists. The Attempt at a Solution First I divided both sides by y^3 and then subtracted t^2 from each as well, -t^2 + y'(y^-3) = 0 then solved, (-t^3)/3 +...
  29. M

    The classic second order differential equation

    If we have \frac{1}{X(x)} \frac{d^2 X}{dx^2}=-κ^2, the literature is saying that the solution must be: e^(±iκx), but am always getting e^(±k^2x). Isn't the approach is to decently integrate twice and then raise the ln by the exponential? Where am I going wrong? Thanks
  30. C

    How to find critical points for a differential equation?

    Homework Statement f(x,y)= 3x5y2 - 30x3y2 + 60xy2 +150 How do I find the critical points where this equation equals zero? Homework Equations None? The Attempt at a Solution I found the partial derivatives, but I'm stuck now and don't know where to go from here. dx = 15x4y2...
  31. F

    MHB Finding region of xy plane for which differential equation has a unique solution

    Determine a region of the xy-plane for which the given differential equation would have a unique solution whose graph passes through a point (x_0, y_0) in the region. x \frac{dy}{dx} = y What does an xy-plane have to do with anything? I looked up the definition of unique solutions and here it...
  32. F

    MHB Making up a differential equation with no real solutions

    Make up a differential equation that does not possesses any real solutions. step 1)consider the definition of solution Any function \phi defined on an interval I and possessing at least n derivatives that are continuous on I which when substituted into an nth-order ordinary differential...
  33. F

    MHB Verifying solution to first order differential equation

    Verify the indicated function y=phi(x) is an explicit solution of the given equation. Consider the phi function as a solution of the differential equation and give at lease one interval I of definition. (y-x)y'=y-x+8 where y=x+4\sqrt{x+2} So the derivative is y'=1+\frac{2}{\sqrt{x+2}} and the...
  34. M

    What kind of differential equation is this?

    Homework Statement It's been a long time since I've done DE's and now I'm stuck with a monster of this form: y'(t) = a*g'(t) + b*g(t) + c*y(t) where g(t) is a known function and a, b and c are constants. What kind of DE is this, and how can I solve for y(t) -- or better yet, what...
  35. N

    Differential equation bifurcation, how to find equilibrium points?

    Homework Statement What are the bifurcation values for the equation: dy/dt = y^3 +ay^2 Homework Equations The Attempt at a Solution Equilibrium solutions: y^3 + ay^2 = 0 ==> y^2 (y + a) = 0 ==> y = 0 (double root), or y = -a. a = 0 is the sole bifurcation point, since...
  36. O

    Differential Equation - Where am I going wrong?

    Homework Statement Using Euler's Method: a) Find the approximate values of the solution of the given initial value problem at t = 0.1, 0.2, 0.3, and 0.4 using the Euler method with h= 0.1 b) Repeat part (a) with h = 0.05. I am doing part (b). The function is y' = 0.5 - t + 2y...
  37. S

    Solving Differential Equations: A Comprehensive Guide

    Hello I seek solutions of differential equation x3y′′(x)+(ax3+bx2+cx+d)y(x)=0 thank you
  38. T

    Substituting w=y' in the Differential Equation

    y'''-5y''+6y'=8+2sinx If I let w=y', would I be able to solve this differential equation? I'm currently stuck and I just want to know if making this substitution is why I am stuck.
  39. R

    Why Does My Differential Equation Solution Differ by a Sign?

    I cannot figure out where I am going wrong. My answer and the textbook answer are different by a negative sign. Can someone review my work and tell me what I am doing wrong?
  40. K

    Need help with a complex differential equation

    Hi guys! I am really stuck at quite a complicated (from my point of view) differential equation. I would really appreciate any hints or suggestions on how to tackle and solve it if it is possible, thanks...
  41. mesa

    Problem solving a separable differential equation for u

    Homework Statement Solve the separable differential equation for u du/dt=e^(5u+2t) Use the following initial condition: u(0)=13 The Attempt at a Solution Honestly I didn't get very far on this one. I took the natural log of both sides, ln du/dt = 5u+2t And now I am stuck. Should I divide...
  42. A

    Find Differential Equation with This Solution

    1. Find a differential equation with the solution y(x) = (x + C)3 The answer cannot contain C) 2. There are no relevant equations. 3. I'm not entirely sure how to do this; I understand that a differential equation has multiple solutions, but for some reason I'm lost on how to find...
  43. N

    Simple differential equation problem on euler's method? HELP

    how do i find the yk value?? the k, tk and mk values makes sense to me Problem #1: http://imageshack.us/photo/my-images/14/fe0x.jpg/ Solution manual: http://imageshack.us/photo/my-images/5/hu55.jpg/
  44. B

    Constructing A Differential Equation From A Solution Set

    Homework Statement y = c1e3xcos(2x)+c2e3xsin(2x)+c3+c4x Homework Equations Differential Equations. The Attempt at a Solution I am having trouble what the roots are for the c3, and c4 parts. I know they are a repeated root, but is it just k= 0? It seems like it would work, but I...
  45. B

    Constructing a differential equation from the solution

    Homework Statement y = c1e3x+c2xe3x+c3e2xsin(x)+c4e2xcos(x) Homework Equations Differential Equations. The Attempt at a Solution I have the roots of k=3, k=3, k=2+i, k=2-i. Now I am just stuck on how to put the roots together to get the original equation. I am just stuck on the complex...
  46. C

    Differential Equation Containing Natural Log of Negative e

    Hi I am working on a problem that ends up having the natural log of a negative e which I'm confused on how to find the explicit solution. The Problem: Find an explicit solution with C. y'-e^{-y}cos(x)=0 My Conclusion: First of all, I'm confused how I should solve this explicitly if I'm...
  47. X

    Differential equation describing RLC cicuit

    Homework Statement Find the input/output differential equation for the LRC circuit in the given figure. The figure is shown in the attachment. Homework Equations V(t) = Ri(t) For inductor, v(t) = Ldi(t)/dt I(t) = 1/L∫v(λ)dλ For capacitor, dv(t)/dt = i(t)/C v(t) = 1/C∫i(λ)dλ...
  48. H

    Convert differential equation to finite difference equation

    I have the differential equation \frac{dM}{dt}=4\pi \rho(r,t)r(t)^2\frac{dr}{dt} which is the first term from M(t)=4\pi\int_0^{r(t)}C(r,t)r(t)^2dr This describes the change in mass (M) of a sphere from a change in radius (r) given a density (rho) that depends on radius and time (t). My...
  49. H

    Differential equation for changing mass of a sphere

    The mass of a sphere with density as a function of radius is M=4\pi \int_0^r\rho(r) r^2dr Lets say the radius increases and decreases as a function of time t. So: M(t)=4\pi \int_{0}^{r(t)}\rho (r) r(t)^2dr I want to know the basic equation describing the mass added or removed...
  50. N

    Use Laplace to solve this differential equation

    Homework Statement Obtain the solution of the differential equation x'' + w2nx = t My use of L refers to the Laplace Homework Equations The Attempt at a Solution L{x'' + w2nx = t} I decided to do the Laplace of each part individually starting with x'' L{x''} =...
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