Differential equations Definition and 999 Threads

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. SrVishi

    Rigorous Differential Equations text

    Hello, I am a math major and I was wondering if you guys knew what would be a good rigorous differential equations text. I really like rigor (like Rudin analysis style rigor or whatnot), instead of the typical books that just focus on the method. I want the proofs and everything. I also would...
  2. S

    Coupled differential equations for charged particles

    Hello, I wanted to study the behaviour of electrons in a spatially bounded system. I want to have a larger number of electrons, but I took 3 to start with and arrived at this system of coupled equations: \begin{align}\begin{bmatrix} \mathbf{\ddot{x_{1}}}\\ \\ \mathbf{\ddot{x_{2}}}\\ \\...
  3. B

    Two differential equations -- Need to find steady state values

    Homework Statement I have found the two equations describing the system. They are. ċ(t) = 1/θ [ f`(k(t)) -(n + δ + β)]c(t) k̇(t) =f(k(t) - (n +d)k(t) - c(t) Plugging in the numbers: ċ(t) = 2 [ 0,25k^(-0,25) -(0,10)]c(t) k̇(t) =k^0,25 - (0,6)k(t) - c(t) Since ċ(t) and k̇(t) =0 in steady...
  4. Prof. 27

    First Order Linear Differential Equation

    Homework Statement dv/dt = 9.8 - 0.196v Set in correct form: dv/dt + 0.196v = 9.8 Since p(t) = 0.196, u(t) the integration factor is given by: u(t) = e∫0.196 dt Multiply each term by u(t) and rearrange: (e∫0.196 dt)(dv/dt) + (0.196)(e∫0.196 dt)(v) = (9.8)(e∫0.196 dt) From now on we will set...
  5. SU403RUNFAST

    Differential equations, orthogonal trajectories

    Homework Statement you are given a family of curves, in this case i was given a bunch of circles x^2+y^2=cx, sketch these curves for c=0,2,4,6, both positive and negative, solve the equation for c and differentiate both sides with respect to x and solve for dy/dx. You obtain an ODE in the form...
  6. twoski

    Differential Equations y" + y = tanx, solve this DE

    Homework Statement 1. y" + y = tanx, solve this DE. 2. dP/dt = P(1 - P) where P = (c1et / 1 + c1et), verify that P is a solution to this DE. 3. Given the pair of functionsx and y, show they solve this system: dx/dt = x + 3y dy/dt = 5x+3y x = e-2t + 3e6t y = -e-2t + 5e6tThe Attempt at a...
  7. Destroxia

    Calculus 3 vs Differential Equations

    My school requires me to take a calc 3 course and differential equations for my major. I am scheduled right now to take differential equations in the summer, and calc 3 in the fall of this year. My school recommends they be concurrent classes, but I'm just too strapped down by my already almost...
  8. mudweez0009

    2nd Order, homogeneous Differential Equation

    Homework Statement Solve d2θ/dη2 + 2η(dθ/dη) = 0, to obtain θ as a function of η, where θ=(T-T0)/(Ts-T0) EDIT: I should add that this is a multi-part problem, and η is given as η=Cxtm. We had to use that to derive the equation in question above.. So I don't know if this is supposed to be...
  9. I

    Laplace Transform of A Second Order ODE

    23.) y'' + 2y' + y = 4e-t; y(0) = 2, y'(0) = -1 Y(s) = [(as + b) y(0) + a y'(0) + F(s)]/(as2 + bs + c) My attempt: a = 1, b = 2, c = 1 F(s) = 4 L{ e-t } = 4/(s+1) (From Laplace Transform Table) Plugging and simplifying: Y(s) = (2s2 + 5s + 7)/[(s + 1)(s2 + 2s + 1) Here is where I get...
  10. B

    What Is the Correct Method to Solve This Differential Equation?

    Homework Statement y3(dy/dx) = (y4 + 1)cosx 2. The attempt at a solution I solved for the homogeneous equation which is y = Ce-sinx Where C is some constant for the particular solution I tried Asinx + Bcosx where A and B are constants but when subbing in it's gets very messy. How should...
  11. S

    Prove cos^2(x)+sin^2(x)=1 using IVP's

    Ok so for this problem I have to use IVPs to prove that cos^2(x)+sin^2(x)=1. I know the end result is suppose to be: du/dt= - v, u(0)=1 dr/dt= u, v(0)=0 but I have no idea how to go about getting to this point.
  12. M

    Derivation of the equations of APF

    I am working to use the artificial potential field method for path planning of mobile robot; actually I found in one of references the following description about this method: the artificial potential field method uses a scalar function called the potential function. This function has two...
  13. G

    Differential equations: applying force to projectile

    Homework Statement The following series of differential equations represents a projectile's path when solved (g=9.81): Modify this series of differential equations to account for an additional force F with vector components a and b acting on the projectile. Here is a sample plot of this...
  14. B

    Engineering Finding the Second Constant in an XRLC Circuit Current Response Problem

    Homework Statement FIND i(t). [/B]Homework Equations VL = L*(di/dt) (Equation 1) IC = C*(dV/dt) (Equation 2) I(t) = e^-9*t*(A*cos(4.359*t) + B*sen(4.359*t)) A (Equation 3) The Attempt at a Solution I can get the exact value of the constant B as shown in the answer if I use -6 after the...
  15. mudweez0009

    Transient heat conduction of a semi-infinite solid

    Homework Statement This is a problem regarding transient heat conduction in an undefined semi-infinite solid, initially at a temperature T0 whose surface temperature is suddenly raised to a new constant level at Ts. I also supplied the problem as an attachment for ease in explaining the...
  16. K

    Poisson PDE in polar coordinates with FDM

    I want to solve a Laplace PDE in a polar coordinate system with finite difference method. and the boundary conditions: Here that I found in the internet: and the analytical result is: The question is how its works? Can I give an example or itd?Thanks
  17. K

    Population Dynamics: Logistic Model (Differential Equations)

    Homework Statement Population Dynamics: Logistic model. Suppose the environmental carrying capacity of the population is 100000 and the growth rate a t=0 is 5%. . If the population starts at 10000, how long does it take for the population to reach half the carrying capacity? dp/dt = A P...
  18. END

    Reducing the function of a function to the ind. variable

    Hey, PF: If I have a function ##f(x)## where ##x## is itself a function of another variable (say time), is the following then true? $$f(x)=f(x(t))=f(t)$$ I ask this because if I have the following system of differential equations $$\frac{dA}{dt}=-Bb$$ $$\frac{dB}{dt}=-Aa$$ where litte ##a##...
  19. U

    Do i need differential equations?

    A friend recently gave me a book on quantum mechanics. It's called Introduction to quantum mechanics. It's by David j griffiths. I am currently taking multivariable calc.I am taking linear algebra next semester. I want study this book, but I am wondering what mathi I need. My friend told me I...
  20. patrickbotros

    Absolute Values in Separable Differential Equations

    When solving a separable differential equation, my textbook says this: ln|v-49|=-t/5+C→ |v-49|=e-t/5+C→ v=49+ce-t/5 What happened to the absolute values? I think it has something to do with the exponential always being positive.
  21. T

    Differential Equations in simple mechanics?

    I came across a few problems in the Kleppner and Kolenkow book in which you must find the force of tension at specific lengths on a rope of mass m. They said you must use differential equations to solve these types of problems. How can you solve and use differential equations like this to get...
  22. B

    Differential equations and escape velocities

    Homework Statement Good day all! I'm stumped on a question: If I fire a bullet straight up what will be the initial velocity such that the bullet doesn't come back down? I need to model a differential equation (it will be first order) some how! Also, Gravity is not constant, but rather, the...
  23. evinda

    MHB Find solution that satifies the differential equations and has a specific form

    Hello! (Smile) I am looking at the following exercise: Let $I=(0,1)$. Find the solution $\phi$ that has a continuous derivative in $\mathbb{R}$ and satisfies : $$y''=0 \text{ in } I \\y''+k^2y=0 \text{ apart from } I, \text{ where } k>0$$ and furthermore $\phi$ has the form...
  24. A

    Applied Differential Equations in physics

    Hi everyone, First of all, this is an awesome place :) I'm looking for a differential equation book, with partial differential equation (and Fourier series solution) that really goes into physics. I'm a 3-year undergrad student in Physics so I already know a little about it. By the way, for now...
  25. M

    Other Looking for a good differential equations text

    Good afternoon, I've been working my way through Serge Lang's series of textbooks, and I recently completed A First Course in Calculus. I'm currently working through the sequel to that book, Multivariable Calculus, and that should keep me tied up for at least two months. Looking ahead...
  26. C

    Linearity in differential equations

    Homework Statement Is the equation (x2sinx + 4y) dx + x dy=0 linear This problem also asks me to solve it, but I don't have a problem with that part. Homework Equations An equation is linear if the function or its derivative are only raised to the first power and not multiplied by each other...
  27. J

    Can Euler-Lagrange Equations Explain Mirages?

    Homework Statement On very hot days there sometimes can be a mirage seen hovering as you drive. Very close to the ground there is a temperature gradient which makes the refraction index rises with the height. Can we explain the mirage with it? Which unit do you need to extremalise? Writer the...
  28. B

    Help Differential Equations types

    Homework Statement Good day all, My professor gave my class a packet of about 40 differential equations. I for the life of me cannot figure out how t solve these last 4! I also have an exam tomorrow morning, and would like finish these last few. I don't need them solved out, I would just...
  29. O

    How to reduce higher order partial differential equations

    Hello guys, I have the system of PDE below and I want to solve it using finite difference method but I think I have to reduce it first to a system of first order PDE. The problem is that I don't know how to reduce this PDE to a first order system. I will appreciate any hints in this regard...
  30. A

    Converge pointwise with full Fourier series

    I am working on a simple PDE problem on full Fourier series like this: Given this piecewise function, ##f(x) = \begin{cases} e^x, &-1 \leq x \leq 0 \\ mx + b, &0 \leq x \leq 1.\\ \end{cases}## Without computing any Fourier coefficients, find any values of ##m## and ##b##, if there is any...
  31. A

    Physics investigation guidance: Vibration of circular plate

    Hi, I have previously made a post in order to gain some insight in my rather out of control project. Long story short I am investigating vibration of a circular plate and its standing waves. After consultation at this forum I have been guided in the direction of acoustics and bessel functions...
  32. L

    Homogenous and linear differential equations

    Homework Statement I am currently in quantum chemistry, and in class one day my professor spent some time talking about Maxwell's equations. I am looking at my notes, trying to piece together Maxwell's equations, differential equations, and the principle of superposition, since this is not in...
  33. O

    Limits of Differential Equations

    Homework Statement I need help finding the limit of the differential equation. (dx/dt) = k(a-x)(b-x) that satisfies x(0)=0 assuming a) 0<a<b and find the limit as t->infinity of X(t) b) 0<a=b and find the limit as t->infinity of X(t) Homework Equations none The Attempt at a Solution I...
  34. D

    A few queries on the variation of parameters method

    I've been reviewing my knowledge on the technique of variation of parameters to solve differential equations and have a couple of queries that I'd like to clear up (particularly for 2nd order inhomogeneous ODEs), if possible. The first is that, given the complementary solution...
  35. I

    MHB Determining Order of Differential Equations

    Hello. I can't seem to remember how to do these kind of problems. I need to determine the order of the differential equations. Can someone show how this is done so that I can understand how to do the rest? $\d{^2y}{x^2}+2\d{y}{x} \d{^3y}{x^3}+x=0$
  36. D

    Applied Physics Differential Equations Problems/Solutions book?

    I want to develop my feel for reading/interpreting differential equations, particularly in Physics applications (for developing models and understanding ones that others have developed). Are there any good books out there that present problems that require deriving governing differential...
  37. electronic engineer

    Third order differential equation

    Hi all, I need to understand these differential equations specially moving from the second order to the third order because i couldn't understand how they got to the result, what was exactly the principle: $$ y'=f(x,y) $$ $$ y''=\frac{df}{dx}(x,y(x)) = f_{x}(x,y) + f_{y}(x,y)y' = f_{x}(x,y) +...
  38. I

    MHB Differential Equations Course: What to Expect & How to Prepare

    Hey MHB. I'm going to be taking a course on differential equations. What can I expect? Is the course harder than Calc 3? Which of the Calc classes is it most similar to? Are there certain topics I should brush up on? I'm really scared about this class and I want to make sure I do well. Thanks :o...
  39. neosoul

    Complex numbers and differential equations for physics

    How relevant is complex analysis to physics? I really want to take differential equations but I would have to change my schedule around way more than I want to. So, would anyone advise a physics major to to take complex analysis? Should I just change my schedule around so I can take differential...
  40. S

    How to extract physical meaning from differential equations

    Homework Statement I am given the following coupled differential equations: \begin{align} (r^2+1)\ddot{θ}+2r\dot{r}\dot{θ} &= u1\\ \ddot{r}-r\dot{θ}^2&=u2 \end{align} together with the following expression for the kinetic energy: \begin{align} T &= 0.5(r^2+1)\dot{θ}^2+0.5\dot{r}^2...
  41. 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...
  42. D

    High order differential equations: undetermined coefficients

    Homework Statement If the method of undetermined coefficients is used to find a particular solution yp (t) to the differential equation y'''-y'=te^(-t)+2cos(t) should have the form: ?Homework EquationsThe Attempt at a Solution LHS r^3-r=0 roots= 0, 1 y_c(t)=c_1e^tRHS te^(-t)+2cos(t)...
  43. 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)...
  44. M

    "State of the Art" in nonlinear differential equations?

    In my introductory ODE class we have focused mostly on linear differential equations. I know that nonlinear differential equations are much harder to solve, and I am wondering what exactly the "state of the art" methods are for dealing with them, or also what recent developments have been made...
  45. M

    Calculus II Differential Equation Question

    Homework Statement We are just starting to learn about basic differential equations in Calc 2 and I learned about separable differential equations but I'm not familiar with this style, here's the question: Given the differential equation of the form ay"+by'+y=0, find constants a and b so that...
  46. B

    Solving Differential Equations: Integrating (y+1)^2/y dy = x^2 ln x dx

    I have differential equations problem, the problem is: ##\frac{(y+1)^2}{y} dy = x^2 \ln x \ dx## Integrating both sides will yield: ##\frac{1}{2} y^2 + 2y + \ln y = \ln x \ . \frac{1}{3}x^3 - \frac{1}{9} x^3 + c## Is this the final solution? If not, what is the final solution?
  47. G

    Help with Differential Equations (First order)

    Homework Statement Solve the given the two equations: xdy + ydx = ydy and (y^2 + 1)dx +(2xy + 1)dy = 0 Homework Equations N/A. The Attempt at a Solution For the first, I can see that solving this with respect to dy/dxmight be a bit tricky. However, if I solve it for dx/dy, things...
  48. _N3WTON_

    Differential Equations with Discontinuous Forcing Functions

    Homework Statement Solve the given initial value problem: y'' + y = u(t-\pi) - u(t-2 \pi) y(0) = 0 y'(0) = 1 Homework EquationsThe Attempt at a Solution First I took the Laplace transform of both sides: \mathcal{L}[y'' + y ] = \mathcal{L}(u(t-\pi)) - \mathcal{L}(u(t-2 \pi)) (s^{2}Y(s)...
  49. _N3WTON_

    Differential Equations with Discontinuous Forcing Functions

    Homework Statement Solve the given initial-value problem. y'' = 1 - u(t-1) y(0) = 0 y'(0) = 0 Homework EquationsThe Attempt at a Solution First I took the Laplace transform of both sides: \mathcal{L}(y'') = \mathcal{L}(1 - u(t-1)) s^{2}Y(s) - sy(0) - y'(0) = \mathcal{L}(1) -...
  50. T

    Solving a System of Differential Equations with Complex Eigenvalues

    1. Homework Statement http://puu.sh/cSK1u/62e2f1c74d.png olve the system: x' = [-4, -4 4, -4] with x(0) = [ 2, 3] Find x1 and x2 and give your solution in real form.2. Homework Equations 3. The Attempt at a Solution Just a note here, I'm basically forced to self-learn this course because...
Back
Top