Differential equation Definition and 1000 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.

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  1. 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''} =...
  2. L

    Can This Complex Differential Equation Be Simplified for Easier Solving?

    Homework Statement Solve the following differential equation: Homework Equations 2 y (y'')^2 + 2 y y''' y' -2 (y')^2 y'' = - (y')^2 The Attempt at a Solution I don't know if the following is useful, but if you divide both sides by y^2, the LHS of the above becomes: (2 y...
  3. L

    Differential equation, Bernoulli's

    Homework Statement Hello everybody :) Now, I have a differential equation to solve. Its a Bernoulli's type of eq. Homework Equations (2x2lny-x)y' = y The Attempt at a Solution I tried of putting it in this way: y'/y = 1/2x2lny-x and then considerin z=1/y, but it doesn't seem...
  4. B

    Differential Equation: Separation of Variables

    Homework Statement dL/dp = L/2, L(0) = 100. Find the solution to the differential equation, subject to the given initial condition. My textbook says the answer is L = 100ep/2, but I don't know how to get that answer (or e for that matter). Homework Equations ? The Attempt at a...
  5. B

    Finding The Solution of A Differential Equation

    Homework Statement y' + 3y = t + exp(2t)Homework Equations The Attempt at a Solution My solution is y = t/3 + exp(-2t) -1/9 + c. The solution in the back of the book is y = cexp(-3t) + (t/3) - (1/9) + exp(-2t). How come the two solutions differ so vastly? I followed the procedure outlined in...
  6. S

    What is the solution to the ODE with p=1/3, f(x)=-3/2x, and g(x)=2/3?

    i tried to solve this question in all the ways i knew but it wouldn't work ..please help xy^2dy/dx + y = x^2 i tried to solve it by using linear first order differential equation technique and also by using different exact and reducable exact differential equaions... help me
  7. B

    Determing If Two Functions Satisfy A Differential Equation

    Homework Statement t^2y" + 5ty' + 4y = 0 Possible solutions: y_1(t) = t^{-2} and y_2(t) = t^{-2} lntHomework Equations The Attempt at a Solution I was able to verify that y_1 was a solution, by the substituting the function, and its derivatives, into the differential equation, which...
  8. B

    Determing Whether A Function Is A Solution To A Differential Equation

    The differential equation: y' -2ty = 1. The possible solution: y=e^{t^2} \int^6_0e^{-s^2}ds + e^{t^2}. For the integral, I employed integration by parts: Let u=e^{-s^2} \rightarrow du = -s2e^{-s^2}ds and Let dv = ds \rightarrow v=s. This lead to: [se^{-s^2}|^t_0 - \int_0^t...
  9. F

    MHB In the simplest qualitative terms what is a differential equation?

    I'm going to be taking a course in differential equations and I'm nervous. From previous calculus courses I know the derivative is the ratio of how one quantity changes with respect to another the integral is the area under the curve So what's a differential equation? According to here "A...
  10. L

    Nonlinear coupled differential equation system - kinetics

    Dear All, Recently, I have measured a series of nonlinear vibrational spectra from which I would like to extract some useful information about kinetics of the exchange process occurring in the studied system. I need to fit my experimental data to kinetic model that is a solution of coupled...
  11. A

    Question on Ordinary Differential Equation (ODE)

    Homework Statement Find the ODE of the following (1) du/dy = -u (2) d^2u/dxdy = -du/dx Homework Equations For question 1, the answer is u= A(x)e^(-y) while for question 2, the answer is u= e^(-y)(B(X) + c(Y)) The Attempt at a Solution I've already solved the question, but...
  12. H

    Modeling a Differential Equation for Air Strut Pressure and Volume

    Hello all, Would love for someone to help me get a solution for this. So, in essence I am trying to model a differential equation for the following scenario: (I am currently achieving this through Excel, but I'm certain a mathematical representation can make things more concise and...
  13. R

    Economics: Elasticity Differential Equation Question

    Hi. Let me preface this by saying that I know nothing about economics. However, I learned a little bit about the concept of price elasticity of demand (that for something really elastic, if price goes up a little, demand will go down a lot), and I came across an equation relating price, demand...
  14. D

    Solving Differential Equation: {x^2} - 3{y^2}+6xy

    Sorry I couldn't think of any more relevant title. Here's the equation: {x^2} - 3{y^2} + 6xy\frac{{dy}}{{dx}} = 0 I'm thinking of rewriting the above to \frac{{dy}}{{dx}} = \frac{{3{y^2} - {x^2}}}{{6xy}} followed by a change of variable u=y/x. But should I rule out the case when either x=0 or...
  15. G

    Ran into this differential equation, got stuck

    Trying to make a calculation and I ran into the following diff equation: r''=a+b/r^2 And I can't seem to remember how one would go about solving it. b = 1355041.84 r(0) = 1400 r(t) = 239.6 r'(o) = 0 r'(t) = 75.2 a = ? I'm specifically looking to find the value of the constant a...
  16. T

    Insanely hard differential equation

    I'm trying to find the equilibrium size of a planetesimal which is made of a material whose density is pressure dependent. (quite a mouthful) I have to solve this differential equation: y(x^2y''+2xy')+2y^2+bxy'=ax^2y^3+(xy')^2+by where a and b are constants.
  17. G

    Differential Equation with Function of Shifted Variable

    Hey guys, I have a question for you... how would one go about solving an equation like this... or this... This came across my mind the other day. I was wondering how to solve that equation if the function is of t-1, instead of t. Obviously, if it was f(t), the solution would be Ce^t, but I was...
  18. R

    A Difficult Differential Equation

    So, as a result of a thought experiment, I've got a differential equation, which I can't solve: R r'' \sin \frac{r}{R} - 2 (r')^2 \cos \frac{r}{R} - R^2 \cos \frac{r}{R} \sin^2 \frac{r}{R} = 0 , R > 0 To make the matters worse, the function r(\varphi) will probably depend on multiple...
  19. N

    Some questions regarding a differential equation

    Hello, I am new here. I hope I am posting my problem at the right place. I need some urgent help regarding the following differential equation: A\frac{d^{2}y}{dx^{2}}+B\frac{dy}{dx}=f(x,\lambda)...(1) where, A and B are constants. x and \lambda are independent. I have solved...
  20. jeffer vitola

    MHB Inexact Differential Equation with 4 Degree

    hello ... I wanted to demonstrate the solution of a differential equation, and as I do not yet know the latex language, then I take pictures of the leaves with my solutions ... the user who wants to give more solutions is well received and is welcome ... att jefferson alexander vitola (Smile)...
  21. MarkFL

    MHB General Solution for $(x^2+y^2)x-y$ Differential Equation

    Find the general solution for: \left(\left(x^2+y^2 \right)x-y \right)\,dx+\left(\left(x^2+y^2 \right)y+x \right)\,dy=0
  22. S

    First-order nonlinear ordinary differential equation

    hii, how to solve this differential equation: A*(dT(x)/dx)(1873.382+2.2111T(x))=90457.5-2.149*10^-10* (T(x))^4 where A is a constant Thank you
  23. J

    Second Order Differential Equation problem

    I have a small problem in solving a Second Order Linear Constant Coefficient Differential Equation. Please see the attached image. I understand upto the point above the arrow. What I don't understand is how he got dy/dx = 9
  24. Albert1

    MHB What is the general solution to the differential equation $xy'-2y =x^2$?

    Please find the general solution of : $xy'-2y =x^2$
  25. FeDeX_LaTeX

    Solving a Differential Equation with a Substitution

    I have the differential equation: 4(2x^2 + xy) \frac{dy}{dx} = 3y^2 + 4xy The only thing I could see working is a substitution, but I can't work out which one to use. I've tried letting v = xy, or v = y/x, but neither of those seem to produce anything useful. Can anyone give me a hint?
  26. K

    Help needed for solving 2nd order differential equation

    can someone help me solve the differential equation that takes the following form? y''+Ay'+By+Cy^2=f(x), y is function of x Thanks a lot!
  27. R

    Differential Equation Question

    Homework Statement Felix, a Red bull skydiver, (mass m) who drops out of a hovering helicopter, falls long enough without a parachute(so the drag force has strenght kv2 ) to reach his first terminal velocity ( denoted v1). When his parachute opens, the air resistance force has a strenght Kv...
  28. 1

    Calculus 2- Differential Equation Mixing Problem

    Homework Statement A tank containing 400 liters of water has 10 kg of salt solute (dissolved salt). Some brine containing 0.03kg/L of salt is then introduced at a rate of 2 L/min. The solution is constantly mixed and evacuated at a rate of 2L/min, such that the volume remains constant. If...
  29. Petrus

    MHB Differential Equation challenge

    Hello MHB, I wanted to post a challange question that is hopefully not really difficult, if the question is not understandable make sure to write it so I can try explain!:)Calculate the Differential equation for y''+2y'=0 that satisfy \lim_{x->\infty}y(x)=1 and y(0)=0 Regards, |\pi\rangle
  30. F

    How Do You Solve This Complex Partial Differential Equation?

    Hi All, Please I need your assistance to solve this PDE below: \frac{\partial^2 X}{\partial t^2} - \frac{\partial^2 X}{\partial z^2} + a(z,t) \frac{\partial X}{\partial t} + b(z,t) \frac{\partial X}{\partial z} +c(z,t) X =\Phi(z,t) With initial and boundary condition...
  31. Petrus

    MHB First order linear differential Equation

    Hello MHB, (x^2+1)y'-2xy=x^2+1 if y(1)=\frac{\pi}{2}What I have done: Divide evrything by x^2+1 and we got y'-\frac{2xy}{x^2+1}=1 we got the integer factor as e^{^-\int\frac{2x}{x^2+1}}= e^{-ln(x^2+1)} Now I get (e^{-ln(x^2+1)}y)'=e^{-ln(x^2+1)} and this lead me to something wrong, I am doing...
  32. S

    Series solution of a differential equation

    Homework Statement Solve the following differential equation by power series and also by an elementary method. Verify that the series solution is the power series expansion of your other solution. y'' = - 4y Homework Equations The Attempt at a Solution y'' = - 4y \\ \frac{d^{2}y}{dx^{2}}...
  33. K

    MHB Help rearranging a linear first order differential equation

    Hi I am trying to solve dy/dx = 3x^2-2x+2+(8/x *y) Can anyone show me how to rearrange to standard form as I am mightly confused :(
  34. U

    Solving this second order linear differential equation

    Homework Statement Find the general solution to the differential equation y'' -16y = 0, where y is a function of x. Give initial conditions that would give a unique solution to the eqution. For the differential equation y'' - k2y = R(x), with k ≠ 0 a real constant, show that it has a...
  35. L

    What is the General Solution to the Differential Equation?

    Hi! I would like to ask what is the general solution of the following differential equation \frac{\partial X_x}{\partial t} = - \frac{\partial X_t}{\partial x} Thank you very much. P.S. If you have some good resiource about this tyoe of equation to recommend please do so.
  36. A

    How to Solve State Homogeneous Differential Equations Using MATLAB?

    can anyone help me with MATLAB code for for solving a state homogeneous differential X=state vector Xdot(t)=AX(t)+BU(t) U is the control input i have the...
  37. J

    Second order differential equation

    Find a solution to the following second order differential equation xy'+y=1/y^2 My Attempt: P= y'= dy/dx x dy/dx + y = 1/y^2 dy/dx + y/x = 1/xy^2 Integrating Factor = e^∫1/x dx = e^lnx y e^lnx=∫ (e^lnx)(1/xy^2) dx
  38. T

    Solution to differential equation

    Obtain the solution to the differential equation: \frac{dy}{dx} = \frac{1+y^2}{1+x^2} Multiple choice answer: a) \frac{Cx}{1-Cx} b) \frac{Cx}{1+Cx} c) \frac{C-x}{1-Cx} d) \frac{1-Cx}{x+C} e) \frac{x+C}{1-Cx} Tried integrating two sides to arrive at arctan y = arctan x + C, but not sure...
  39. F

    Differential equation dimension analysis

    A differential equation of solitary wave oscillons is defined by, $$ \Delta S -S +S^3=0 $$ **How can we write this equation as,** \begin{equation} \langle(\vec{\nabla}S)^2\rangle+\langle S^2\rangle-\langle S^4\rangle=0 \tag{1} \end{equation} where $\langle f\rangle:=\int d^Dx f(x)$...
  40. P

    Uniqueness - exact differential equation

    Hi folks! This one got me in doubts... Homework Statement Solve IVP (Initial Value Problem): (2xy+sin(x))dx+(x^{2}+1)dy=0, y(0)=2 Is the solution unique? Motivate why! Homework Equations Relevant equations for solving the exact equation... The Attempt at a Solution I can...
  41. S

    Can I separate a differential equation?

    Given the following differential equation x*x''+(x')^2+y*y''+(y')^2=C where C is a constant and all differentiation is with respect to time Can i equal the first and second parts of the equation into different constants and solve separately?, meaning solving the system...
  42. A

    Linear system of equations for a differential equation

    Solve the equation m\frac{d^{2}x}{dt^{2}} + c\frac{dx}{dt} + kx = (ax + b)^{2} + c^{2} for the constants m, c, k The right hand side a, b, and c are arbitrary digits. For me they are a = 2, b = 3, and c = 8. The problem recommends creating a linear system of equations for me to solve. This...
  43. P

    First order linear differential equation

    Homework Statement dy/dt = k*y*ln(y/M), where M and k are constants. Show that y = Meaekt satisfies the above equation for any constant a. Homework Equations y' = ky y = P0ekt The Attempt at a Solution Taking the derivative of y, I get: (Meaekt)*(aekt)*k which is, ky*aekt ..and I'm...
  44. I

    Nonlinear second order differential equation

    Hello! It is the first time that i write on this forum. I'm doing a PhD but i can't solve this equation: it's a non-linear second order differential equation. ay''+b|y|y'+cy+dx=0 Some ideas?
  45. B

    Find a particular solution for a non-homogeneous differential equation

    Find a particular solution for the following equation: y"+2y'+y=12.5e-t I'm not sure on which method to use. Here's my attempt using the undetermined coefficients method: →y"+2y'+y=12.5e-t r2+2r+1=0 r=-1 *not even sure if this part is useful →yp=e-t yp'=-e-t yp"=e-t...
  46. 1

    Differential Equation Involving Heaviside function

    Homework Statement y''+y= H(3pi)(t) y(0)=1 y'(0)=0 H(3pi)(t) means the heaviside function such that it is 1 above 3pi and 0 otherwise. Homework Equations The Attempt at a Solution Taking the laplace of both sides, I have that: L(y) = (L(H(3pi)(t))+s) / (s^2 + 1)...
  47. Sudharaka

    MHB Solving a Riccati Differential Equation

    Hi everyone, :) Continuing from http://www.mathhelpboards.com/f17/method-reduction-order-variation-parameters-4574/ thread, my friend gave me another question. He wants to check his work specifically on parts a) and d). So I will only focus on those parts. Question Summary: Given the Riccati...
  48. S

    Second Order Linear Differential Equation (Non-constant coefficients)

    Homework Statement Find the general solution for the following differential equation: y'' + x(y')^2 = 0. Homework Equations Integration, differentiation... The Attempt at a Solution Usually for these sort of DE you could use the substitution v(x) = y'(x) and this would simplify...
  49. R

    On the regular singularities of a second order differential equation

    Homework Statement The only singularities of the differential equation y''+p(x)y'+q(x)y=0 are regular singularities at x=1 of exponents \alpha and \alpha', and at x=-1 of exponents \beta and \beta', the point at infinity being an ordinary point. Prove that \beta=-\alpha and \beta'=-\alpha'...
  50. B

    Solving a Bernoulli differential equation

    Hello! I'm stuck at the moment with this differential equation. I've been trying to use the method for solving these equations, but my answer is not correct according to my book. Could anyone please explain what I'm doing wrong? Thanks!
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