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. moriheru

    Differential equation practice problems book?

    Salve, Can anybody recommend a book which has lots of differential practice problems?Or a similar book for Laplace transforms? No explanations just the answers and the problems,simply for practicing. Thanks for any help.
  2. D

    Differential equation for unbalanced rotating force

    Homework Statement ##M\ddot{y} + k_{eq}y = me\omega^2\sin(\omega t)## What is ##m##? Homework EquationsThe Attempt at a Solution In the ODE above, ##M## is the total mass of the problem, correct? For instance, if we had a cantilever beam, ##M = m_b + m_m(0.23)## where ##m_b## is the mass of...
  3. D

    Differential Equations System Solutions

    Homework Statement Consider the initial value problem for the system of first-order differential equations y_1' = -2y_2+1, y_1(0)=2 y_2' = -8y_1+2, y_2(0)=-1 If the matrix [ 0 -2 -8 0 ] has eigenvalues and eigenvectors L_1= -4 V_1= [ 1...
  4. D

    2nd order differential equation: undetermined coefficients

    Homework Statement y''-y=t-4e^(-t)Homework Equations method of undetermined coefficients The Attempt at a Solution solving for characteristic equation first y''-y=0 r^2-1=0 c_1e^(-t)+c_2e^(t) RHS particular solution t-4e^(-t) y_p(t)= At+B+Ce^(-t) y_pt'(t)=A-Ce^(-t) y_p''(t)=Ce^(-t)...
  5. J

    Spherical coordinates of Partial Differential Equation

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  6. Sneakatone

    Differential equation with laplace transform and springs

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  7. M

    Calculus II Differential Equation Question

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  8. 2

    Setting up a differential equation (buoyancy)

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  9. R

    Differential equation/characteristic equation

    Any help is appreciated 1.)----Find the Character equation for the diff equation d^2y/dx^2-4dy/dx+3y=0 with initial conditions y(0)=0 and y'(0)=12 find the solution y(t) (this is what I have gotten so far on this part) p^2+4p+3=0 then (p-1)(p-3)=0 so p1=1 and p2=3? not really sure...
  10. S

    Can someone help with my differential equation involving power series?

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  11. Soumalya

    TextBooks for Some Topics in Mathematics

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  12. TheTank

    Difficult differential equation

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  13. DontDeimos

    Differential equation for a rocket

    Homework Statement A rocket through interstellar dust, no gravity. Solve for v with respect to mass. k is a constant. Drag = -bv Homework Equations [/B] The Attempt at a Solution To start: It should become a separable differential equation, but I am having a lot of trouble solving it...
  14. bearcharge

    Associated Legendre differential equation involved in solving spin function?

    Amazed by the closeness of equations for orbital angular momentum L and spin angular momentum S, I can't help asking is associated Legendre differential equation involved in solving spin function? I only heard that spin naturally comes from treatment of quantum mechanics with relativity theory...
  15. D

    Differential equations - backwards problem

    Homework Statement If y=y(t) is the solution of the initial value problem { y'+(2t+1)y=2cos(t) y(0)=2 then y''(0)=? it is a multiple choice practice problem with choices y''(0)=2 y''(0)=-2 y''(0)=4 y''(0)=0 y''(0)=-4Homework EquationsThe Attempt at a Solution Im really not sure how to go...
  16. D

    Differential Equations: Bernoulli Equation

    Homework Statement Find the general solution: y'-3y=(y^2) Homework EquationsThe Attempt at a Solution divide both sides by y^2 y'(y^-2) -3(y^-1) = 1 we know v=y^(n-1) v=y^-1 v'=d/dx(y^-1) v'=-(y^-2) y' plug it back into y'(y^-2) -3(y^-1) = 1 -v'-3v=1 this is where I think I am making a...
  17. C

    Can you prove a differential equation has no analytical solution?

    Can you prove a differential equation has no analytical solution? Teach me ,please! Thank you a lot!
  18. Q

    Solving a matrix differential equation in Mathematica

    I want to solve a differential equation of the form d/dt (R) = B(t) where R and B are two complex matrices, with time dependent functions as their coefficients. I want to solve these 3x3 differential equations (ones on the diagonal are actually coupled) and to obtain a matrix form for R. So...
  19. B

    Suspension system, linear differential equation

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  20. M

    Differential Equation with Noise term

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  21. B

    Separation of Variables in 2 Dimensions: A Laplace Equation Primer

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  22. M

    Finding vC(t) for a Critical Damping RLC Circuit

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  23. Z

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  24. J

    Simple Partial Differential Equation

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  25. fisher garry

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  26. C

    Numerical solution of a differential equation with time dependent terms

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  27. P

    Differential equation for an LCR circuit

    Homework Statement Find the differential equation that Q(t) satisfies. 2. Relevent equations Kirchoffs loop law and voltage across a capacitor, resistor and inductor. The Attempt at a Solution [/B] So I'm thinking, by Kirchoffs voltage rule, that the sum of the voltages in this circuit...
  28. SteliosVas

    Solve a Simple Differential Equation | Step-by-Step Guide

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  29. Y

    Euler's Method to approximate a second order Differential Equation

    Homework Statement y'' + 4y' + 4y = 0 ---- y(0) = 1, y'(0) = 5 Find the exact solution of the differential equation. Use the exact solution and Euler's Method to compute Euler's Approximation for time t = 0 to t = 5 using a step h=0.05. Plot Euler's & Exact vs. t and plot Error vs. t. Then...
  30. T

    Separable differential equation

    Homework Statement 2y * y'(t) = 3t^2 such that y(0) = 9 Homework Equations g(y)y'(t) = h(t) The Attempt at a Solution So I have done many of these seperable ones in homework that did not require a parameter so now I got lost. This is what I did;[/B] Integral of 2y dy = integral of 3t^2...
  31. P

    Problem with a Differential in a Non Inertial System.

    A ball of mass 'm' is inside of a tube that rotates in a horizontal plane around the vertical axis (Drawing a circunference). Attached to the ball (inside of the tube) there is a massless, inextensible rope that goes to the midpoint of the circle described by the rotating tube. The other end of...
  32. Daaavde

    Proof of Hamiltonian equations

    So, I should prove that: - \frac{\partial H}{\partial q_i} = \dot{p_i} And it is shown that: - \frac{\partial H}{\partial q_i} = - p_j \frac{\partial \dot{q_j}}{\partial q_i} + \frac{\partial \dot{q_j}}{\partial q_i} \frac{\partial L}{\partial \dot{q_j}} + \frac{\partial L}{\partial q_i} =...
  33. rakeru

    Is the Derivative of a Function a Differential Equation?

    Is the derivative of a function a differential equation? I guess it would be because it involves a derivative, right? Would the solution to the equation just be the original function? Is solving a differential equation just another way of integrating? Like with finding solutions of separable...
  34. S

    Anyone help with this differential equation

    from the attached file I get something like the answer as y(x) = e^(e^2x(x/3*(sin(3x)+1/9cos(3x)+c) Im not sure if this is right can you have an answer exponetial of an exponential??
  35. trash

    Maxima and minima of differential equation

    Homework Statement Consider the differential equation y'=x-y^2. Find maxima, minima and critical points; show that for every solution f=f(t) exists T\geq 0 such that f(t)< \sqrt{T}\;\forall t > T Homework Equations The Riccati equation: y'=a(x)y^2+b(x)y+c(x) The Bernoulli equation...
  36. teroenza

    Series solution to nonlinear differential equation

    Homework Statement By truncating the differential equation below at n=12, derive the form of the solution, obtaining expressions for all the ancoefficients in terms of the parameter \lambda .Homework Equations The ODE is: \frac{\mathrm{d^2}\phi }{\mathrm{d} x^2} = \frac{\phi^{3/2}}{x^{1/2}}...
  37. F

    MHB Help on a 2nd Order Linear Differential Equation

    Hi, I need help solving this ODE. I know the answer is a Bessel function but I need help on the process of getting there. Initial conditions y(0)=1 and y'(0)=0 xy''+y'+xy=0 I have made it this far... ${x}^{2}*\sum_{n=0}^{inf} [n(n+1)*{C}_{n+2}+(n+1)*{C}_{n+1}+{C}_{n}]*{x}^{n}=0$
  38. _N3WTON_

    Homogenous Differential Equation Solution

    Homework Statement Solve the differential equation: \frac{dy}{dx} = x + y Homework Equations Integrating factor equationThe Attempt at a Solution Ok, I am aware that in order to solve this equation, I need to make a substitution: v = x + y However, at this point I am unsure about what to...
  39. matt_crouch

    First order partial differential equation

    How do I go about solving a differential equation of the form \partial_{x}F_{x}(x,y) + \partial_{y}F_{y}(x,y) = g(x,y) Where g(x,y) is a known function and I wish to solve for F. I thought i could apply the method of characteristics but the characteristic equation is dependent on coefficients...
  40. F

    MHB First Order Nonlinear Differential Equation

    Hi, I need help solving this ode, when I try to solve it I end up with a big crazy answer and I believe it should be simpler. (dy/dx)^2=((ay^4)/2)-(a+1)y^2+1 y(0)=0, y'(0)=1 and a is within [0,1]
  41. F

    MHB Need Help Solving a 2nd Order Nonlinear Differential Equation

    Solve the 2nd order nonlinear differential equation, with initial conditions y(0)=0 and y'(0)=1 y''=2ay^3-(a+1)y with a within [0,1] I am pretty much lost on how to go about solving this. It would be greatly appreciated if someone could point me in the right direction on this. Thanks!
  42. D

    Differential equation that describes the voltage with respect to time

    In class we discussed RLC circuits in series. My homework problem, however, has me analyzing an RLC circuit in series. a) Write the differential equation that describes the voltage with respect to time. b) Find the natural frequency of oscillation and the damping constant of the circuit...
  43. C

    Finding a differential equation of a growing object

    Homework Statement Mass , r, of a round object grows in proportions with its SA. The density of the object is constant. Find the differential equation for the change in 'r' of the object over time. Arbitrary constants can be combined but final answer must depend on only r and constants...
  44. R

    Periodic Solution to Differential Equation

    For each epsilon greater than 0, show that the differential equation x'=x^2-1-cos(t)-epsilon has at least one periodic solution with 0 less than x(t) less than or equal to (2+epsilon)^1/2
  45. S

    Why Does Substitution Fail in Solving Differential Equations?

    Homework Statement I want to solve number 13. I have my work on the attached file. Homework Equations The Attempt at a Solution I've tried to solve the system by elimination and substitution. I keep failing again and again
  46. C

    How Do You Solve This Tricky Differential Equation Problem?

    Homework Statement y'sin(2t) = 2(y+cos(t)) y(\frac{∏}{4}) = 0 Homework Equations \frac{dy}{dx} + p(x)y = q(x) y = \frac{\int u(x) q(x) dx + C}{u(x)} where u(x) = exp(\int p(x)dx) The Attempt at a Solution I've set the equation in the form above, simplified the RHS and solved for...
  47. J

    2nd Order Nonlinear Differential Equation

    Hi everyone, I need some help to solve this differential equation. The question states "Use the perturbation or multiple scale method to find the third-order approximate solution for the following system: diff(x(t), t, t)+w^2*x(t)*(1+epsilon*x(t)^2) = 0 " Currently, I am still...
  48. C

    Need Help With a Separable Differential Equation

    Hello. I need some help solving a differential equation. I think where I'm going wrong is integrating one side via partial fractions, but I'm not quite sure where my mistake is. Using Wolfram, I found the correct answer, which is below. Thanks. Homework Statement Solve the following initial...
  49. L

    Solving partial differential equation

    Hi all , I would like to solve the following partial differential equation. (∂α/∂t)=G[sub(α)] *(a'-a) I attached the equation and solution here as an image. I don't know how it was derived. I hope someone can help me
  50. F

    MHB Definition of differential equation

    My textbook defines differential equation as Could someone explain what is meant by the part in parenthesis "dependent variables"? I don't see the difference.
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