What is Differential equations: Definition and 999 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. B

    System of Partial Differential Equations

    System of PDEs--Heat Equation For Two Objects Hello everyone, Before is a system of partial differential equations; to be specific, it is this system: \frac{\partial U_A }{\partial t} = - \frac{k_B}{k_A} \alpha_A \left( \frac{\partial^2 U_B}{\partial x^2} + \frac{\partial^2 U_B}{\partial...
  2. A

    Homogeneous differential equations

    Is this a homogeneous DE? 3y'''' + 21y'' + y' + 6y = 0 So... since a(n-1)y''' is missing, would this still by definition be a homogeneous differential equation?
  3. A

    Laplace transform for set of differential equations

    I have a set of differential equations with the basic form: dy_n/dt = t*(a_(n-1)*y_(n-1)+b(n+1)*y_(n+1)-2c_n*y_n) So the time depence is a simple factor in front of the coefficient matrix. Does this set of differential equations have closed form solutions?
  4. A

    Practice problems for differential equations

    I can't really find much differential equations problems workbook. I really want one but resources are limited. Can you please send amazon links and etc. that I can order a differential equations problems workbook? Also, it would be greatly appreciated that the workbook at least covers topics...
  5. N

    Symbolic solve coupled second order differential equations

    Dear all, I have posted a similar question in another forum and the general consensus seems to suggest that it is not possible to symbolic solve a system of coupled second order different equation with damping (dissipation) and driving forces. However, I have found in many papers and books...
  6. E

    MATLAB Coupled 1st order differential equations in matlab

    Hello everyone I just want to ask if anyone could help me or at least tell me if it is possible to solve the couple equations using matlab I saw ode45 but that is for one equation only Thank uou
  7. F

    Differential equations true/false question

    Homework Statement If y1 and y2 are solutions to y"-y = 0 then c1y1+c2y2 represent all solutions to the differential equation for all scalars c1 and c2 Homework Equations The Attempt at a Solution Basically, my TA's solutions to his worksheet said it was true, and I'm not sure if...
  8. D

    Differential equations (application)

    Homework Statement hi, i have difficulties in this question... can you teach me how to get the ans please... i don't have the ans . this involved differential equations Homework Equations The Attempt at a Solution
  9. S

    Differential equations, zero state response

    Homework Statement Find the zero input and zero state response for the following system y''(t) + 3y'(t) + 2y(t) = 2 x'(t) - x(t-1) where x(t) = (2e^-t)*u(t) U(t) is the step function Homework Equations Y = Yh + Yp Y = Yzsr + Yzir The Attempt at a Solution I can't find...
  10. A

    System of linear differential equations

    For a system of linear differential equations with constant coefficients with known initial conditions an analytical solution can be found. I however have a system of linear differential equations, where the coefficients are timedependent with the dependence of the coefficients being...
  11. K

    Solving Differential Equations: A Guide to 2 ln |20-2h| = kt + c

    -2 ln |20-2h| = kt + c kinda lost. solving (i) would be enough for me
  12. JasonHathaway

    Differential equations and Cramer's rule

    Hi everyone, I'm taking the Differential Equations for the first time, and I want to know the most helpful textbook for the subject. We had the following example: Find the differential equation which its general solution is: y=C_{1}+C_{2} x+x^{2} Solution: y^{'}=0+C x+2x y^{''}=0+0+2...
  13. MexChemE

    Partial differential equations?

    Hello, PF! As I was reading my P-Chem textbook, I noticed most thermodynamic equations involve partial derivatives, like these ones: C_V = {\left( \frac {\partial E}{\partial T} \right )}_V {\left( \frac {\partial H}{\partial T} \right )}_P = {\left( \frac {\partial E}{\partial T} \right )}_P +...
  14. W

    Exploring Solutions for Differential Equations and Simple Harmonic Motion

    Simple harmonic motion: ##y'= -z,~z'= f(y)##the modified explicit equation are$$y'=-z+\frac {1}{2} hf(y)$$$$y'=f(y)+\frac {1}{2} hf_y z$$ and deduce that the coresponding approximate solution lie on the family of curves $$2F(y)-hf(y)y+z^2=\textrm{constant}$$where ##f_y= f(y)##. What are the...
  15. W

    Differential equations

    Show that the nonlinear oscillator $$y" + f(y) =0$$ is equivalent to the system $$y'= -z $$, $$z'= f(y)$$ and that the solutions of the system lie on the family of curves $$2F(y)+ z^2 = constant $$ where $$F_y= f(y)$$. verify that if $$f(y)=y$$ the curves are circle. => nonlinear oscillator...
  16. F

    Differential Equations - first order

    Homework Statement a function has the feature that at any point, the product of its gradient and the x-cordinate is equal to the square root of the y-cordinate multiplied by 5. Part A: Write out a differential equation that describes the function Part B: If the curve passes through the point...
  17. PsychonautQQ

    Differential Equations Method of Undetermined Coefficients

    Homework Statement consider y'' + 2y' - 3y = 1 + xe^x, find the particular solution The Attempt at a Solution so f(x) = 1 + xe^x f'(x) =e^x + xe^x f''(x) = 2e^x + xe^x so it looks like my particular solution is going to have a constant term, an e^x term and an xe^x term, so I can...
  18. A

    Coupled differential equations

    A bit related to my other topic but here goes: I have the set of differential equations: x1' = a-ax1-ax2 x2' = b-bx1-bx2 What is the solution to such a set? I could google systems of differential equations, but that turns up large texts on the theory. I just wonna know in a few lines...
  19. D

    Mathematics differential equations

    Homework Statement hi, all… can anyone help me with this question? i got stucked here? can you figure out which part contains mistake or post the full solution here? thanks in advance! http://i.imgur.com/RZpquyd.jpg?1 http://imgur.com/RZpquyd&Eoaa0mm#0 Homework Equations The...
  20. z.js

    Can some one teach me Differential Equations?

    I would like to learn this. Can someone teach me? The things like δy2/δ2x and solving for x and y. :rolleyes:
  21. R

    MHB Advanced Numerical solution of differential equations

    i) IF $\frac{dy}{dt} = - \frac{∂H}{∂z}, \frac{dz}{dt}= \frac{∂H}{∂y}$ where H is a function of $y$ and $z$, show that $H(y,z)$ is constant in time. ii) Take a $H(y,z) =Ay^2 + 2Hyz + Bz^2$ where $A,B,H$ are constants and show that solutions of the system lie on ellipses. iii) Apply the...
  22. L

    Is This the Correct Method for Solving Differential Equations on Midterms?

    Hi, this question came up in my midterm and I was hoping to know if this is the correct method or answer.
  23. L

    MHB Differential Equations by separation of variables

    Can someone please help me to calculate the following using separation of variables: dy/dx = x*(1 - y^2)^(1/2) to that the solution is in the form: y =
  24. K

    MHB Algebraic manipulations for system of differential equations

    I have a problem I would like some guidance on. I need to find the values of $k$ for which $x^2+ky^2$ is a Liapunov function for the system $$\dot{x}=-x+y-x^2-y^2+xy^2, \dot{y}=-y+xy-y^2-x^2y$$ **My attempt:** $$\dot{V} = \frac{\partial V}{\partial x} \times \frac{dx}{dt} + \frac{\partial...
  25. M

    Torrcellis law differential equations

    1. The problem statement, all variables and given/known This is a group project for differential equations. I ended up without a group, lucky me. I've been trying to work through this on my own but I am stuck. Sorry about the pictures, typing it all out would of taken ages...
  26. alexsylvanus

    Separable Differential Equations Using Initial Values

    AP Physics student here, I'm working on a problem that takes into account air resistance, where something is thrown up at initial velocity v_0, and the drag force is proportional to the velocity, so, \vec{F_{drag}}=-k\vec{v}. Using Newtons second law and making up positive, down negative, you...
  27. B

    Solving Differential Equations with Initial Conditions

    Hello Homework Statement Find the general solution of the following diff erential equations. In each case if y = 2 when x = 1, fi nd y when x = 3. x \frac{dy}{dx} = \frac{1}{y} + y Homework Equations The Attempt at a Solution x \frac{dy}{dx} = \frac{1}{y} + y x ...
  28. B

    Understanding Differential Equations

    Hello, these are the first differential equations I've tried to solve... Homework Statement Find the general solution of the following differential equations. In each case if y = 2 when x = 1 find y when x = 3. 2x \frac{dy}{dx} = 3 Homework Equations The Attempt at a Solution 2x...
  29. L

    Differential Equations help

    Hello all, I am currently having trouble with this Differential Equations problem. Let x = F(t) be the general solution of x'=P(t)x+g(t), and let x=V(t) be some particular solution of the same system. By considering the difference F(t)−V(t), show that F(t)=U(t)+V(t), where U(t) is the general...
  30. E

    Differential Equations Rate At Which Chemical Amount Changes

    Homework Statement Suppose that a large mixing tank initially holds 300 gallons of water in which 50 pounds of salt have been dissolved. Pure water is pumped into the tank at a rate of 3 gal/min, and when the solution is well stirred, it is then pumped out at the same rate. Determine...
  31. A

    Find the differential equation or system of differential equations

    Find the differential equation or system of differential equations *** Find the differential equation or system of differential equations assoicated with the following flows a) ##\phi_t (x) = \frac{x}{\sqrt{1-2x^2t}} ## on ##{\mathbb R} ## b) ##\phi_t (x,y) = (xe^t, \frac{y}{1-y^t}) ## on...
  32. D

    First order differential equations

    Homework Statement Consider the first order differential equation \frac{dx(t)}{dt} + ax(t) = f(t), x(0) = x_{0}, t\geq0 Suppose the "input signal" f(t)=e^{-t}, t\geq0 . (a) Find the solution to the equation. Find a condition on the parameter a so that the solution of the (forced) system...
  33. K

    Differential Equations: Separable Equations

    Homework Statement Solve the equation dy/dx = x/(y^2√(1+x)) Homework Equations The Attempt at a Solution I separated them: y^2 dy = dx/√(1+x) I then integrated the dy side, I got (1/3)y^3 + C. I am stuck at integrating the dx side. Thanks in advance!
  34. I

    Find a solution for differential equations

    I have to solve the following differential equation (that I found in an article): \frac{1}{r^{5}}\partial_{r}(r^{5}\partial_{r}h(r)) -E\frac{h(r)}{r^{2}} = - \frac{C}{r^{5}}\delta(r-r_{0}) where E and C are two constants. The authors of the article first find a solution of the previous...
  35. L

    Open Loop Transfer Function from Differential Equations

    Homework Statement Generate an open loop u(t) and simulate. Plot x(t) and y(t) \dot{x} = Vcos(θ) \dot{y} = Vsin(θ) \dot{θ} = u I am given initial values. All are 0 except for \dot{x}(0) = V. Homework Equations Laplace Transform Tables The Attempt at a Solution I think I...
  36. S

    Solve system of two differential equations

    Homework Statement Solve the system ##\dot{x}=-2x+y+5sint## and ##\dot{y}=x-2y+3## for ##x(0)=1## and ##y(0)=1## Homework Equations The Attempt at a Solution Well, I tried expressing x from the second equation, derived it and inserted it in the first equation but this process...
  37. M

    Differential equations assignment T6

    Hi! I would like to ask anyone with some spare time to check my assignment questions. Last time I was asked to post one task at a time so I will. Thank you in advance for your time. Task 6: In a galvanometer the deflection θ satisfies the diffrential equation: d2θ/dt2+2(dθ/dt)+θ=4...
  38. M

    Differential equations assignment T5

    Hi! I would like to ask anyone with some spare time to check my assignment questions. Last time I was asked to post one task at a time so I will. Thank you in advance for your time. Task 5: Find the particular solution of the following differential equations: a) 12(d2y/dx2)-3y=0...
  39. M

    Differential equations assignment T3

    Hi! I would like to ask anyone with some spare time to check my assignment questions. Last time I was asked to post one task at a time so I will. Thank you in advance for your time. Task 3: A capacitor C is charged by applying a steady voltage E through a resistance R. The p.d. between the...
  40. M

    Differential equations assignment T2

    Hi! I would like to ask anyone with some spare time to check my assignment questions. Last time I was asked to post one task at a time so I will. Thank you in advance for your time. Task 2: Determine the equation of the curve which satisfies the differential equation...
  41. M

    Differential equations assignment T1

    Hi! I would like to ask anyone with some spare time to check my assignment questions. Last time I was asked to post one task at a time so I will. Thank you in advance for your time. Task 1: Solve the differential equation: x(dy/dx)+x2=5 given that y=2.5 when x=1 Solution: x(dy/dx)+x2=5...
  42. F

    Differential equations and series

    1. 1.Homework Statement Use power series to evaluate intial value problem y''-xy'-2y=4x^2 y(0)=1 and y'(0)=1 Homework EquationsThe Attempt at a Solution I got the series Cnx^n and took its derivative and plugged it into the formula and I got but when I factor out x^n and set that...
  43. R

    Ordinary differential equations

    Consider the first order differential equation dy/dt = f(t,y)= -16t^3y^2, with the inital condition y(0)=1. Estimate the lipschitz derivative for the differential equation by substituting the exact solution into ∂f/∂y. =I found the exact solution by using the separable of variable and...
  44. D

    Differential equations involving the function composition

    I have not met differential equations involving the composition functions (also not much literature on it). Assume we know the form of g=g(x), and need to solve the following differential equation, finding f=f(x): (g∘f)f'=g Where g∘f=g(f(x)). Does anybody have a strategy for solving...
  45. H

    Help putting differential equations into matrix form

    Homework Statement Hello, I am trying to put the following equations into matrix form in order to solve the system. If anyone could please explain to me how to do it or show me an example it would be awesome. All material given in question: For the system of inhomogeneous differential...
  46. G

    Is this true about differential equations?

    If a_3(x)y'''+a_2(x) y''+a_1(x) y'+a_0(x)y=f(x) is an ODE with particular solution y_{p1} and a_3(x)y'''+a_2(x) y''+a_1(x) y'+a_0(x)y=g(x) is an ODE with particular solution y_{p2}, then the ODE a_3(x)y'''+a_2(x) y''+a_1(x) y'+a_0(x)y=f(x)+g(x) has the particular solution y_{p1}+y_{p2}.
  47. D

    Generalized eigenvectors and differential equations

    Let A be an 3x3 matrix such that A\mathbf{v_1}=\mathbf{v_1}+\mathbf{v_2}, A\mathbf{v_2}=\mathbf{v_2}+\mathbf{v_3}, A\mathbf{v_3}=\mathbf{v_3} where \mathbf{v_3} \neq \mathbb{0}. Let B=S^{-1}AS where S is another 3x3 matrix. (i) Find the general solution of \dot{\mathbf{x}}=B\mathbf{x}. (ii)...
  48. T

    Solving System of Two Differential Equations

    Homework Statement Find General Solution of the Following System (2D+5)x - (2D+3)y = t (D-2)x + (D+2)y = 0 https://dl.dropboxusercontent.com/u/32294083/Emath/New%20Doc%203_1.jpg Using the Quadratic Formula I get nothing so I am not sure what the complementary solution is...
  49. I

    Dividing differential equations

    The point of my question is that when we divide a differential equation by a function or variable we result in different solution (not always). Take the example: ydx+ydy=0, constaint: xy=a By substituting x with y/a and after some manipulations we arrive to (-a/y)dy+ydy=0 and on...
  50. F

    Differential equations incongruecy

    Homework Statement I am going to copy-paste this text that my friend made (because we both have the same doubt and we don't know to work around it. This is a long post, so warning): "I'm currently unsure of how these two problems work. I've tried working at them in different ways but i don't...
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