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. M

    So the equation becomes u' + (1/x)u = x

    Homework Statement equation (1) y' + (ylny)/x = xy we set y(x)= eu(x) Equation (1 ) proposed becomes an equation 1st order linear differential type : equation (2) u ' + p( x) u = q ( x) a) Find the equation ( 2) using the change of variable proposed above. Homework EquationsThe Attempt at a...
  2. Remixex

    Question about mg on oscilations course

    On a test our teacher asked about a system composed of (string -> mass -> string -> mass) hanging, that began to oscillate up and down. We all considered weight (mg) when applying Newton's second law to find the associated differential equation. When we met our teacher again he said that we...
  3. M

    AP Physics differential equation

    Homework Statement A block of mass m, which has an initial velocity v0 at time t = 0, slides on a horizontal surface. If the sliding friction force f exerted on the block by the surface is directly proportional to its velocity (that is, f = -kv), A) Write a differential equation for the...
  4. B

    How do I solve these coupled Differential Equation?

    Homework Statement dNa/dt = -Na/Ta where Na is the function and Ta is the constant dNb/dt = Na/Ta - Nb/Tb where Nb is the function and Tb is the constant Homework Equations My Prof said Nb(t) has the form Nb(t) = Cexp(-t/Ta) + Dexp(-t/Tb) The Attempt at a Solution I know the first equation...
  5. M

    Find value of K for differential equation to be exact

    Homework Statement Hello everyone, I need to find K for the following differential equation to be exact. (y3+kxy4-2*x)dx +(3xy2 +20x2y3)dy=0 Homework EquationsThe Attempt at a Solution (y3+kxy4-2*x)dx +(3xy2 +20x2y3)dy=0 dM/dy = d/dy(y3+kxy4-2*x) = 4*k*y3*x+3*y2 dN/dx =d/dx(3xy2 +20x2y3) =...
  6. J

    Keep getting the wrong answer in this lengthy Laplace problem

    Homework Statement y'' + 4y = 8t^2 if 0 < t < 5, and 0 if t > 5; y(1) = 1 + cos(2), y'(1) = 4 - 2sin(2). Use the Laplace transform to find y. Homework Equations t-shift, s-shift, unit step function. The Attempt at a Solution I have been trying to solve it for hours, but keep getting the wrong...
  7. M

    Homogeneous differential equation - serious help

    Homework Statement I need to resolve this with v = y/x dy/dx= (3y2-x2)/(2xy) Homework EquationsThe Attempt at a Solution dy/dx= (3y2-x2)/(2xy) dy/dx= 3y2/2xy -x2/2xy dy/dx = 3y/2x -x/2y dy/dx = 3y/2x - 1/2y/x dy/dx = 3/2 *v - 1/2*v F(v) = 3/2 *v - 1/2*v is that good so far ?
  8. wololo

    Circular motion with friction differential equation

    Homework Statement Homework Equations F=ma ac=v^2/r f=uN v=v0+at w=v/r The Attempt at a Solution v=v0+at v=vo+umv^2/r v^2(u/r)-v+vo=0 I don't see what differential equation i could use since the speed is dependent on the friction (equal to friction coeff times centripetal force) which in...
  9. nuuskur

    Bernoulli differential equation, Cauchy problem

    Homework Statement Observe a Cauchy problem \begin{cases}y' + p(x)y =q(x)y^n\\ y(x_0) = y_0\end{cases} Assume ##p(x), q(x)## are continuous for some ##(a,b)\subseteq\mathbb{R}## Verify the equation has a solution and determine the condition for there to be exactly one solution. Homework...
  10. R

    Engineering LC circuit (Differential Equation)

    Homework Statement An inductor with value L and a capacitor with value C are connected in series to a power source. At time t, the voltage of the power source (i.e. the voltage across both the inductor and capacitor) is given by ## v(t)=Asin(\frac{2t}{\sqrt{LC}}) ##. If the voltage across the...
  11. B

    Solving a problem regarding Existence theorem.

    Homework Statement Given the equation dy/dx = y^4 - x^4, y(0) = 7, determine whether the existence/uniqueness theorem implies that the given initial value problem has a unique solution. Homework Equations Existence/Uniqueness Theorem The Attempt at a Solution To my understanding, you must...
  12. Shahrokh

    Coupled differential equation with boundary conditions

    Hi, I have two coupled differential equations d^2 phi(z)/dz^2=lambda*phi(z)*(phi(z)^2+psi(z)^2-sigma^2) d^2 psi(z)/dz^2=lambda*psi(z)*(phi(z)^2+psi(z)^2-sigma^2+epsilon/lambda) where lambda, epsilon and sigma are arbitrary constants. The equation subject to the bellow boundary conditions...
  13. W

    First order integro differential equation

    Can anyone help me to solve a differential equation? I want to solve ∂v(p,t)/∂t=-p^2 v(p,t)-sqrt(2/pi)∫v(p,t)[1-δ(t)R(t)]dp+sqrt(2/pi)[δ(t)R^2(t) C] with initial data v(p,0)=0 where C is constant and the integration from zero to infinty Any suggestion please? Solution by volterra integral...
  14. A

    Solve second order nonlinear differential equation

    how do you solve this equation? y´´ + k/(y^2) = 0 ? I got it from applying Newton's 2nd law of motion to an object falling from space to Earth only affected by gravitational force. Thank you!
  15. R

    Verifying a solution of the diffusion equation

    This is my attempt at the solution. I have been told that the given function is a solution. I just need to prove it. As you can see that I am stuck. What am I doing wrong?
  16. MidgetDwarf

    Solving a Differential Equation by Separation of Variables

    Solve the given differential equation by separation of variables. (dy/dx)= (xy+3x-y-3)/(xy-2x+4y-8) First, I noticed when i divided both sides by the left hand side and multiplied both sides by dx, nothing canceled or seemed to work. I got to thinking. on the right hand side I preformed long...
  17. H

    MHB Exponential Growth and Differential Equation

    a) On January 1 2000, the park estimated that they had 500 deer on their land. Two years later, they estimated that there were 550 deer on the land. Assume that the number of deer was changing exponentially, i.e. P(t)=ae^bt where P is the number of deer at year t, and a and b are parameters.Find...
  18. Mark Brewer

    Ordinary Differential Equation Problem

    Homework Statement dy/dx = 4e-xcosxThe Attempt at a Solution [/B] I've divided dx to both sides, and now have dy = 4e-xcosx dx I've then started to use intergration by parts to the right side with u = 4e-x and dv = cosx dx Leaving y = 4e-xsinx - ∫ -4e-xsinx dx Once again I used intergration...
  19. H

    Seek power series solutions of the given differential equation

    I know there are a number of ways to do this problem, to increment the series etc. but, would someone please be able to explain how they get the answers for this problem simply and easily ? Thanks! A screen shot is attached
  20. W

    Is this differential equation exact?

    Homework Statement Identify the following differential equation as linear, separable, exact, or a combination of the three. $$1 + \frac{1+x}{y}\frac{dy}{dx} = 0$$ Homework Equations Start with ##F(x,y)=C## ##\displaystyle \frac{d}{dx}(F(x,y)) = \frac{d}{dx} (C)## ##\displaystyle...
  21. R

    Higher order differential equation

    Solve y'' - 2y' + 2y = ## e^x tanx ## What concept should we use if we know only solving first order differential equation?
  22. J

    Setting up differential equations for voltage

    Let V1 be the voltage across C1 and V2 be the voltage across C2. I want to solve for V1 and V2 as a function of time. My idea was to use dV1/dt=I1/C1 and dV2/dt=I2/C2. Then using circuit rules i can express I1 and I2 as functions of V1 and V2 and substitute them into the previous diff eqs...
  23. D

    Analytical solution to mx"(t)+b(x'(t))x'(t)+k(p)x(t)=0

    Hello guys! I have following differential equation mx"(t)+b(x'(t))x'(t)+k(p)x(t)=0. As can be seen, "attenuation term" is dependent of velocity x'(t). Also stiffness term k(p) is dependent of term p, which is p=k(p)x(t)/A. In this equation A is constant and k(p) means, of course, same term as...
  24. SSGD

    What is this differential equation? I'm going crazy

    I have been working on a math problem and I keep getting the some type of PDEs. x*dU/dx+y*dU/dy = 0 x*dU/dx+y*dU/dy+z*dU/dz = 0 ... x1*dU/dx1+x2*dU/dx2+x3*dU/dx3 + ... + xn*dU/dxn= 0 dU/dxi is the partial derivative with respect to the ith variable. Does anyone know about this type of PDE...
  25. kostoglotov

    Question concerning 2nd order homogeneous linear diff eqs

    Homework Statement Regarding the case where the auxillary (characteristic) equation has complex roots, we solve the quadratic in the usual way using i to get the general solution y(x) = e^{\alpha x}\left(C_1 \cos{\beta x} + i C_2 \sin{\beta x}\right) And the textbook shows y(x) = e^{\alpha...
  26. Destroxia

    Step Function IVP Differential Equation w/ Laplace Transform

    Homework Statement (didn't know how to make piecewise function so I took screenshot) Homework EquationsThe Attempt at a Solution My issue here with this problem is that I have absolutely no idea where to start... I have read through the textbook numerous times, and searched all over the...
  27. Destroxia

    Higher Order Differential Equation

    Homework Statement ##y^{(4)} + y = 0, y(0)=0, y'(0)=0,y''(0)=-1,y'''(0)=0## My issue with this equation is not with the steps, I don't believe but the solving of the IVP, the derivatives of my solution end up being close to 32 terms long, and I was wondering if there is any shorter method I...
  28. Destroxia

    Finding series solution for the differential equation

    Homework Statement y'' - xy' + xy = 0 around x0=0 Find a solution to the 2nd order differential equation using the series solution method.Homework Equations Assume some function y(x)= ∑an(x-x0)n exists that is a solution to the above differential equation.The Attempt at a Solution How...
  29. Remixex

    Hooke's and Newton's law to find Second order ODE

    Homework Statement A weight of 8 pounds extends a spring 2 feet. It's assumed that the damping force that acts on the system is equal (number-wise) to alpha times the speed of the weight. Determine the value of alpha > zero so x(t) is critically damped. Determine x(t) if the weight is liberated...
  30. E

    Hyperbolic partial differential equation

    What is the general solution of the following hyperbolic partial differential equation: The head (h) at a specified distance (x) is a sort of a damping function in the form: Where, a, b, c and d are constants. And the derivatives are with respect to t (time) and x (distance). Thanks in advance.
  31. Destroxia

    Differential Equation, Substitution/Homogeneous

    Homework Statement Solve the differential equation: Here is the books answer to the problem: Homework Equations This is a substitution/homogeneous first order differential equation, which can be converted into a separable differential equation. The Attempt at a Solution Where am I...
  32. W

    Can Poker Chip Arrangements Be Optimized for Maximum Profit?

    Hey everyone. I was pondering how best to optimize a chip arrangement for a poker game. This is the scenario I've thought up: There are 4 denominations of colored chips with a set value. White (W) = 0.05 Red (R) = 0.25 Blue (B) = 1.00 Green (G) = 5.00 A player wants to purchase 40 dollars...
  33. evinda

    MHB How do we get to this differential equation?

    Hello! (Wave) If we have the initial value problem $$h''(t)=-\frac{R^2 g}{(h(t)+R)^2} \\ h(0)=0, h'(0)=V$$ we have the fundamental units: $T,L$ such that $T=[t], L=[h], L=[R], LT^{-1}=[V], LT^{-2}=[g]$ and we get the independent dimensionless quantities $\pi_1=\frac{h}{R}...
  34. evinda

    MHB General solution of differential equation

    Hello! (Wave) $$(1-x^2)y''-2xy'+p(p+1)y=0, p \in \mathbb{R} \text{ constant } \\ -1 < x<1$$ At the interval $(-1,1)$ the above differential equation can be written equivalently $$y''+p(x)y'+q(x)y=0, -1<x<1 \text{ where } \\p(x)=\frac{-2x}{1-x^2} \\ q(x)= \frac{p(p+1)}{1-x^2}$$ $p,q$ can be...
  35. LeDragonian

    How Is Displacement Solved in This Differential Equation?

    It has been a while since I was involved with my differential equations. I am a mech student. I was trying out a sample problem from the dynamics book and came upon this equation. k = some constant of proportionality for a spring pushing back a spring mounted slider. (1/k) arcsin(ks/v_0) = t...
  36. evinda

    MHB Non-linear differential equation

    Hello! (Wave) I want to find the solution $\psi$ of the non-linear differential equation $y'=1+y^2$ that satisfies the condition $\psi(0)=0$. (Notice that the solution $\psi$ exists only for $- \frac{\pi}{2}< x < \frac{\pi}{2}$) We notice that: $(tan^{-1})'(x)=\frac{1}{1+x^2} (\star) \left(...
  37. M

    Explaining a physical model (differential equation)

    Homework Statement Check on labb41.jpeg (I am not used to formatting my mathematical equations on the web, I can only format in my local text-editing programs). Homework Equations Again, check on labb41.jpeg The Attempt at a Solution I have plot the solution to U(I) as according to the...
  38. 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...
  39. D

    Differential equation of frictional force

    A question from a classical mechanics past paper described a particle of mass ##m## that had a pair of horizontal identical springs of spring constant ##k## attached on either side and that the mass is free to move horizontally. The mass is also placed on a table that gives rise to an...
  40. S

    General differential equation solution for Kepler Problem

    To be honest, I don't know any physics. I am a high school student who has taken high school physics, but America's education system isn't known for teaching much more than Newton's laws. I have, however, taken Multivariable/Vector calculus, so I have a decent math background. I was wondering...
  41. gonadas91

    Solve 1st Order ODE from Transcendental Equation

    It is a general doubt about the following equation: Imagine I want to calculate an unknown function y(x), and my starting equation is of the type y(x)^{2}=\frac{1}{x^{2}Log^{2}(A(x)y(x)^{2})} , then, am I allowed to start with the equation y(x)=\frac{1}{xLog(A(x)y(x)^{2})} and...
  42. W

    Separable partial differential equation

    Homework Statement I have two equations. cos(θ)wφ + sin(θ)wφ = 0 (1) And ## \frac{w_φ}{r}## + ∂wφ/∂r = 0 (2) Find wφ, which is a function of both r and theta. Homework EquationsThe Attempt at a Solution I end up with two equations, having integrated. wφ=## \frac{A}{sinθ}## from (1)...
  43. Prof. 27

    Linear Ordinary Differential Equation: Definition

    Homework Statement The website says this: "It is Linear when the variable (and its derivatives) has no exponent or other function put on it. So no y2, y3, √y, sin(y), ln(y) etc, just plain y (or whatever the variable is). More formally a Linear Differential Equation is in the form: dy/dx +...
  44. R

    Finding Time Taken for Particle to Travel 99 Meters Using Differential Equations

    Homework Statement A particle moves in a straight line with velocity given by ## dx/dt = x +1 ## ( x being distance described). The time taken by the particle to describe 99 meters is? Homework Equations NA The Attempt at a Solution Getting ## ln(x+1) = t + C## How to determine the constant...
  45. M

    Differential Equation - Series - Recurrence Relation

    1. (16+x2)-xy'+32y=0 Seek a power series solution for the given differential equation about the given point x0 find the recurrence relation. So I used y=∑Anxn , found y' and y'' then I substituted it into the original equation, distributed, made all x to the n power equal to xn, made the...
  46. G

    How do I solve this stochastic differential equation?

    Homework Statement I am trying to solve this \begin{align} d X_t = - b^2 X_t (1 - X_t)^2 dt + b \sqrt{1 - X_t^2} dW_t \end{align} where $b$ is a constant. Note that I have the answer here and can provide it if necessary. But I want to know how one would come up with it. Homework EquationsThe...
  47. K

    Derive gen sol of non-homogeneous DEs through linear algebra

    Hello, I noticed that the solution of a homogeneous linear second order DE can be interpreted as the kernel of a linear transformation. It can also be easily shown that the general solution, Ygeneral, of a nonhomogenous DE is given by: Ygeneral = Yhomogeneous + Yparticular My question: Is it...
  48. grandpa2390

    How Do Imaginary Numbers Affect the Inverse Laplace Transform Calculation?

    Homework Statement find the inverse laplace transform. 1/(s^3 + 7s) Homework Equations sin(kt) = k/(s^2 + k^2) cos(kt) = s/(s^2 + k^2) The Attempt at a Solution so this one is different from all the others I have done because it involves an imaginary number and I am not sure what the rule...
  49. M

    Complicated (for me) differential equation problem

    1. I am supposed to find dx and dy. I think I am missing a step or a general idea. I spent quite some time figuring out what rules should I use and the only sequence I can think of is quotient rule and chain on the (x2 +y2)1/2 term. The answer that I find is ((xy(3x2+2y2))/(x2+y2)3/2 . On the...
  50. K

    How to solve highly oscillating differential equation

    The equation looks like: ##x''(t)+b x'(t)+cx(t)+ d x^3(t)=0##. This is the motion of a particle in a potential ##cx^2/2+d x^4/4## with friction force ## b x'##. In my case, the friction term is very small and the particle will oscillate billions of times before the magnitude decreases...
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