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. B

    Implicit Solution To Differential Equation

    Homework Statement Verify that the indicated expression is an implicit solution of the given first order differential equation. Find at least one explicit solution in each case. Give an interval I of definition of each solution. The differential equation is: \displaystyle \frac{dX}{dt} = (X...
  2. J

    Solve for the initial value of the differential equation

    Homework Statement Solve the initial value problem: y+(3x-xy+2)dy/dx = 0 , y(1)=1 I couldn't separate with y as a dependent variable, so I made x the dependent variable and I get this: dx/dy= x(1-2/y)-(2/y), in linear standard form: dx/dy+(3/y - 1)x = -2/y. Homework Equations...
  3. A

    Determine for the flow of a differential equation

    In each of the following cases, we define a function : ##\phi##: ##{\mathbb R} \times {\mathbb R}^3 \rightarrow {\mathbb R}^3 ## . Determine in each case whether this function could be the flow of a differential equation, and write down the differential equation. (a) ##\phi_t(\vec{x}) =...
  4. C

    First order non-linear differential equation

    Homework Statement Hello, I was given an extension problem in a Dynamics lecture today and am struggling to solve it. It is a simple scenario: a particle of mass m is accelerating due to Galilean gravity, but is subject to a resistive force that is non-linear in the velocity of the particle...
  5. L

    First-Order Linear Differential Equation

    Homework Statement A 400-gallon tank initially contains 200 gallons of water containing 2 parts per billion by weight of dioxin, an extremely potent carcinogen. Suppose water containing 5 parts per billion flows into the top of the tank at a rate of 4 gallons per minute. The water in the tank...
  6. G

    Differential Equation, Solve for function

    Homework Statement Given \frac{dx}{dt} + ax = Asin(ωt), x(0) = b Solve for x(t) Homework Equations The Attempt at a Solution I take the Laplace transform of both sides and get sX(s) - x(0) + aX(s) = \frac{Aω}{s^{2} + ω^{2}} X(s) = \frac{b}{s + a} + \frac{Aω}{(s^{2} + ω^{2})(s+1)} The...
  7. F

    How can I solve this differential equation for quadratic resistance?

    I'm solving a differential equation to do with quadratic resistance and it seems to be acting very strangely - I get the opposite sign of answer than I should. If anybody could have a quick look through that would be much appreciated. For a particle moving downward and taking positive upwards...
  8. J

    Differential equation to homogenous linear equation.

    can this equation y, = ycot(x) + sin(x) be reduced to a homogenous linear format? If yes, how? I tried the usual y=xv and the x=X+h, y=Y+k but doesn't seem to be working. Any ideas? Thanks just realized its in the form of dx/dy+Px=Q so I solved it by multiplying on B.S. by e∫Pdx and the...
  9. C

    Rodent Population as a differential equation.

    Homework Statement https://www.physicsforums.com/attachment.php?attachmentid=65556&stc=1&d=1389570667 For those who can not see the screen shot here is the question... Suppose the population P of rodents satisfies the diff eq dP/dt = kP^2. Initially there are P(0) = 2 rodents, and their number...
  10. C

    Water drains from a tank. Write a differential equation.

    2. Water drains out of an inverted conical tank at a rate proportional to the depth y of water in the tank. Write a differential equation for y as a function of time. My answer was dy/dt = ky. This was from a weekly homework set where there were only 5 problems. I feel like I am missing...
  11. S

    Differential Equation -> Behaviour near these singular points

    Differential Equation ---> Behaviour near these singular points Homework Statement Problem & Questions: (a) Determine the two singular points x_1 < x_2 of the differential equation (x^2 – 4) y'' + (2 – x) y' + (x^2 + 4x + 4) y = 0 (b) Which of the following statements correctly describes...
  12. T

    MHB Non-linear first order differential equation question

    Kindly solve it and i need help to understand NON LINEAR Questions
  13. K

    MHB Homogeneous, linear, first-order, ordinary differential equation mistake

    I am trying to solve a homogeneous, first-order, linear, ordinary differential equation but am running into what I am sure is the wrong answer. However I can't identify what is wrong with my working?! $$\frac{dy}{dx}=\frac{-x+y}{x+y}=\frac{1-\frac{x}{y}}{1+\frac{x}{y}}.$$ Let $z=x/y$, so that...
  14. B

    How to solve this nonlinear differential equation

    dy/dx=2x+y^2 By the way, methods of solving linear differential equation are useless, such as integrating factor and Bernoulli method.
  15. vanceEE

    Solving a Second Order Differential Equation: $$xy''-y'=3x^2$$

    $$ xy'' - y' = 3x^{2} $$ $$ y' = p $$ $$ y'' = p' $$ $$ xp' - p =3x^{2} $$ $$ p' - \frac{1}{x}p = 3x $$ after multiplying by the integrating factor we get.. $$ \frac{1}{x}p' - \frac{1}{x^{2}}p =3 $$ so $$ [\frac{1}{x}p]' = 3? $$ I know that these two below are equal, but can someone please show...
  16. P

    Calculating Velocity of a Rock Dropped into a Hole Drilled Through the Earth

    Homework Statement Inside the earth, the force of gravity is proportional to the distance from the center. If a hole is drilled through the Earth from pole to pole and a rock is dropped in the hole, with what velocity will it reach the center? The Attempt at a Solution I think that the...
  17. W

    Second Order Differential Equation to show s = ut +(1/2)at^2

    Homework Statement If d^2s/dt^2 = a, given that ds/dt = u and s = 0, when t = 0, where a, u are constants show that s = ut + .5at^2 2. The attempt at a solution du/dt = a cross multiplying and then integrating and we get u = at ds/dt = at cross multiply and...
  18. ChrisVer

    Series Solution to Differential Equation

    I have to solve the differential equation y''+(1-t) y' + y= sin(2t) can someone judge this? How could I continue it? y=\sum_{n=0}^{∞}{a_{n} t^{n}} y'=\sum_{n=1}^{∞}{a_{n} n t^{n-1}} y''=\sum_{n=2}^{∞}{a_{n} n(n-1) t^{n-2}} sin(2t)=\sum_{n=0}^{∞}{\frac{2^{2n}}{2n!} t^{2n}} y''+(1-t) y'...
  19. bhanesh

    Linearity of differential equation

    Friends I have one doubt Below given equation is linear or non linear :)
  20. 5

    Second order non-linear differential equation involving log

    EDIT: my problem is solved, thank you to those who helped Homework Statement Solve: x y^{\prime \prime} = y^{\prime} \log (\frac{y^{\prime}}{x}) Note: This is the first part of an undergraduate applications course in differential equations. We were taught to solve second order...
  21. M

    What is the Solution for Differential Equation Task 7?

    Task 7 Show that y=(1/4)tsin2t satisfies equation d2y/dt2+4y=cos2t Find the general solution and deduce the solution which satisfies y(0)=0 and y'(0)=0. What happens as t increases? Solution In the end I stay with: y=Acos2t+Bsin2t+(1/4)tsin2t...
  22. B

    Finding the general solution of the given differential equation

    y"+2y'+y=2e^-t I tried to find the solution for this nonhomogenous diff. Equation but i could not. First i took a function Y(t)=Ae^-t but i was getting 0=2e^-t. To get rid of that i took another y'+y=2e^-t and found the solution y=2te^-t + ce^-t. Noticed that first part of this finding is...
  23. L

    Differential equation of motion

    Homework Statement Find the maximal ground reaction force for the limiting case where the speed at initial contact is equal 0 (Vo=0) expressing it as a mulitple of the body weight (mg) ω = √k/m Homework Equations Y1(t) = A sinωt + B cos ωt + g/ω^2 The Attempt at a Solution I...
  24. M

    Laplace Differential Equation of a Half-Annulus

    Here is the DE: Δu(r,θ)=0, 1 ≤ r ≤ 2, 0 ≤ θ ≤ pi and here are the Boundary Conditions: u(1,θ)=sin(θ), u(2,θ)=0, u(r,0)=0, u(r,pi)=0 Based on the Boundary Conditions I believe this is half of an annulus. Using the 2D Laplace equation for polar coordinates, find the solution u(r,θ). I've...
  25. P

    Find the Fourier series solution to the differential equation

    Find the Fourier series solution to the differential equation x"+x=t It's given that x(0)=x(1)=0 So, I'm trying to find a Fourier serie to x(t) and f(t)=t, and I'm know it must a serie of sin... So here's my question...the limits of integration to the Bn, how do I define them? Will...
  26. K

    Assistance with Third Order Differential Equation

    Homework Statement Hi, Just wondering if anyone knows how to solve the following as I am not sure where to start at all: y''' + 8y = xsin(2x) Any help would be great. Homework Equations The Attempt at a Solution I'm thinking solving the homogeneous DE y'''+8y = 0 and...
  27. C

    Change of variables in a differential equation in Maple

    Homework Statement Consider the differential equation zZ'' + Z' + γ2 Z = 0, where Z = Z(z). Use the change of variables x = √(z/b) with b a constant to obtain the differential equation Z'' + (1/x)Z' + α2Z = 0, where Z = Z(x) and α= 2γ √b Homework Equations Maple commands The Attempt at a...
  28. TheFerruccio

    Y''+xy'-y=0 differential equation

    Homework Statement Given y_1=x is a solution, solve the differential equation Homework Equations y''+xy'-y=0 The Attempt at a Solution Since I am given y_1=x (is there a hotkey for adding TeX tags so I don't have to manually type these tags over and over? So tedious.) then I...
  29. U

    Differential Equation for deceleration of a bullet

    Homework Statement A bullet of mass m strikes an amor plate with initial velocity v0. As the bullet burrows into the plate, its motion is impeded by a frictional force which is directly proportional to the bullet's velocity. There are no other forces acting on the bullet. -Use Newton's Second...
  30. K

    Solving a Differential Equation with Boundary Conditions

    What is the answer of this differential equation. ((d^2) r)/((ds)^2) +(m/(r^2)) -(nr/3)=0 the boundary conditions (i) r=a when s=0 and (ii) dr/ds =0 when r=b. m and n are constants.
  31. R

    Comp Sci Fortran Programming for order differential equation

    Consider the first order differential equation dy/dy = f(t,y) = -16 t^3 y^2 with initial condition y(0)=1 Using second order Adams-Bashforth method, write a Fortran programming to generate an approximate solution to the problem. please forgive me for not trying because I really...
  32. N

    Solving the differential equation m^2 = 0?

    I have a problem that starts with the equation: r\frac{d^2u}{dr^2} + \frac{du}{dr} = 0 The solution I'm looking at says to do a substitution, letting u = r^m, which after differentiation and simplification results in: m^2 = 0 Up until this point, I understand (well, actually I don't...
  33. S

    Solve differential equation if you now two solutions

    Homework Statement Let ##y_1(x)= e^x## and ##y_2=x^2+1+e^x## solve the differential equation ##y^{'}+b(x)y=c(x)##. Find the overall solution of this differential equation. Homework Equations The Attempt at a Solution The overall solution => ##y=y_H+y_P## I don't know the english expression...
  34. I

    Solving 2nd Order Differential Equation with Initial Conditions

    Mod note: Reinstated problem after poster deleted it. Homework Statement Just wondering if I did this correctly: ##y''+4y'+4y=e^{x}## and initial conditions ##y(0)=0; y'(0)=1## Homework Equations The Attempt at a Solution So I found the characteristic equation to be...
  35. M

    Solving Differential Equations: Examples and Step-by-Step Solutions

    Homework Statement 1st problem - is this correctly done? \frac{dy}{dx} = (##x^2## - 1) ##y^2## , y(0) = 1 2nd problem - I really need help with this one. xy' - y = ##3x^2## , y(1) = 1 The Attempt at a Solution 1st problem: \frac{dy}{dx} = (##x^2## - 1) ##y^2## , y(0) = 1...
  36. J

    Second order differential equation - Drawing circuit

    Homework Statement "Solve differential equation y''+y=2*cos x. Draw circuit of the equation and think about the strange behavior of the current." The attempt at a solution I was able to solve the equation, but I have no idea how to draw circuit about it, we haven't gone through this...
  37. J

    Difficult differential equation

    I have the following differential equation which I want to solve for y as a function of x \frac{dy}{dx}=\frac{C_{1}\left(C_{5}y+C_{6}\right)^{2}}{C_{2}\left(C_{3}y+C_{4}\right)-C_{7}\left(C_{5}y+C_{6}\right)^{6}} where C_{1},C_{2},C_{3},C_{4},C_{5},C_{6},C_{7} are constants. Can anyone...
  38. G

    Solving Differential Equation: Boy & Girl's Meeting Time

    Homework Statement A boy is on a boat, at a distance H from the shore, when he sees a girl (at the point on the shore where the distance is measured) running with a constant velocity u parallel to the shore. At that time, he moves towards her, with a speed v, in such a way, that the point of...
  39. M

    MHB Show that the differential equation has no solution that satisfies g(0) = 1

    hello! I am facing some difficulties at the following exercise. "show that g(x)=x \cdot F(x) , where F(x)=\int_{0}^{x} {s(x)}dt , s(x)=\frac{sin(x)}{x} , satisfies the diffential equation xy'(x)-y(x)=xsin(x) , x ε R, and find all the solutions in this space. Show that the differential...
  40. M

    Is This Differential Equation Exact or Solvable by an Integrating Factor?

    Homework Statement . Solve the differential equation: ##(3x^2-y^2)dy-2xydx=0##. The attempt at a solution. I thought this was an exact differential equation. If I call ##M(x,y)=-2xy## and ##N(x,y)=3x^2-y^2##, then the ODE is an exact differential equation if and only if ##\frac{\partial...
  41. O

    Understanding the General Solution of a Differential Equation

    Hello - I asked a similar question before, but it was not resolved for me, and the person who answered was rude, so I did not continue the conversation. I read this here: http://tutorial.math.lamar.edu/Classes/DE/SecondOrderConcepts.aspx "If y_1(t) and y_2(t)are two solutions to a...
  42. A

    Higher Order Differential Equation: Substitution

    Homework Statement Solve x^{2}\times y'' - 4 \times x \times y' + 6 \times y = 0 for y(x) by first using the substitution v = ln(x) to obtain an equation involving y, dy/dv, d^2y/dv^2 and no x. Solve for y(v), then return to y(x). Homework Equations NA The Attempt at a Solution I know how...
  43. iVenky

    How to solve delay differential equation

    Is there some systematic procedure to solve delay differential equation ? Here's one equation that I would like to solve \large \frac{1}{ \omega } \frac{dV_0(t)}{dt} = V_i(t) - \frac{V_o(t-T_d)}{k} where Td is the delay Thanks
  44. N

    Analytical solution of nonlinear ordinary differential equation

    Dear All, I have following first order nonlinear ordinary differential and i was wondering if you can suggest some method by which either i can get an exact solution or approaximate and converging perturbative solution. \frac{dx}{dt} = 2Wx + 2xy - 4x^{3}\frac{dy}{dt} = \gamma \, (x^{2} -...
  45. F

    Statistical Differential Equation

    Stochastic Differential Equation Hi there, I am trying to solve (analytically) a stochastic differential equation of the form: \frac{d^2}{dt^2}x +\left(k(t)+\delta k \ t\right)x)=0 Here \delta k is a random (gaussian) white noise. Note, that in the differential equation it is multiplied...
  46. A

    Solving a simple differential equation

    I have had this question on my mind for a long time When we solve a differential equation like this \frac{dT}{dx}=0 Do we do this ? \int\frac{dT}{dx}dx=\int0dx\int dT=\int0dxT =c_1 Because if we were to separate variables this doesn't work, we're just integrating both sides in respect...
  47. F

    Differential equation - Reduction of order

    Hi, i have to reduce the order of a 2nd order differential equation, to solve it with a numerical method. The equation is: \ddot{r}+a\dot{r}+\frac{b}{r^{2}}=0 with a,b\geq0 I tried to reduce it substituting \dot{r}=v, but i don't know what to do with the term \frac{b}{r^{2}} ...
  48. C

    Deriving a differential equation for car motion

    I'm looking at this scenario where a car is moving and then shifts into neutral. Knowing the initial velocity, how can I derive a differential equation? I know the air drag and the frictional force... are there any other forces, like gravity, that should be included to make it realistic? I...
  49. L

    Second Order Nonhomogenous Differential Equation

    Hello everyone, I'm having trouble understanding the solutions to DE's of the form: ay''+by'+cy=f(t) We've gone over them in class, I've talked with my friends, and it just doesn't make any sense to me. I was wondering if anyone on here would help me understand the solutions, it would be...
  50. F

    Solving differential equation with a step impulse.

    Solving differential with a step impulse. Hi, I have problem I know I should be able to do but I've been stuck on it for a while. Just looking to be pointed in the right direction. (dq^2/d^2t) + 2*ζ*ω*dq/dt + (ω^2)*q = u(t)/L Where u(t) is a step impulse, q is the charge through an...
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