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. K

    I Definition of order of a partial differential equation

    How is the order of a partial differential equation defined? This is said to be first order: ##\frac{d}{d t}\left(\frac{\partial L}{\partial s_{i}}\right)-\frac{\partial L}{\partial q_{i}}=0## And this second order :##\frac{d}{d t}\left(\frac{\partial L}{\partial...
  2. M

    MHB How to solve this partial differential equation

    What the hell is this and is it solvable?
  3. A

    Solving a first order differential equation with initial conditions

    Hello! Consider this ODE; $$ x' = sin(t) (x+2) $$ with initial conditions x(0) = 1; Now I've solved it and according to wolfram alpha it is correct (I got the homogenous and the particular solution) $$ x = c * e^{-cos(t)} -2 $$ and now I wanted to plug in the initial conditions and this is...
  4. W

    I Nonlinear differential equation

    hi, i am working on nonlinear differential equation- i know rules which decide the equation to be nonlinear - but i want an answer by which i can satisfy a lay man that why the word nonlinear is used. it is easy to explain nonlinearity in case of simple equation i.e when output is not...
  5. A

    Solving a mixing problem with a differential equation

    Hello! First I tried modelling it like most mixing problems. $$ \frac{dA}{dt} = rate coming in - rate coming out $$ where dA is the volume and dt is the time rate coming in/out can be describe as; contrencation * flow rate. Now if we plug that all on $$ \frac{dA}{dt} = 35 * 0 -...
  6. J

    Solving this Differential Equation using Convolution

    $s=c_1*\exp(-c_2*|(t)|)*r(t)$ But how can I solve $c_1+c_2$ ?
  7. L

    I Solve second order linear differential equation

    Consider the second order linear ODE with parameters ##a, b##: $$ xy'' + (b-x)y' - ay = 0 $$ By considering the series solution ##y=\sum c_mx^m##, I have obtained two solutions of the following form: $$ \begin{aligned} y_1 &= M(x, a, b) \\ y_2 &= x^{1-b}M(x, a-b+1, 2-b) \\ \end{aligned} $$...
  8. stevendaryl

    A Differential equation for affine parameter

    Suppose you have a smooth parametrized path through spacetime ##x^\mu(s)##. If the path is always spacelike or always timelike (meaning that ##g_{\mu \nu} \dfrac{dx^\mu}{ds} \dfrac{dx^\nu}{ds}## always has the same sign, and is never zero), then you can define a smooth function of ##s##...
  9. H

    MHB Application of Linear differential equation in solving problems

    A rumour spreads through a university with a population 1000 students at a rate proportional to the product of those who have heard the rumour and those who have not.If 5 student leaders initiated the rumours and 10 students are aware of the rumour after one day:- i)How many students will be...
  10. A

    Homogenous solution of a differential equation

    Hello ! I need to solve this diffrential equation. $$ y^{(4)} + 2y'' + y = 0 $$ First I wanted to find the homogenous solution,so I built the characteristic polynomial ( not sure if u say it so in english as well).I did that like this $$\lambda^4 +2\lambda^2+1 = 0 $$. The solutins should be...
  11. alan123hk

    I How do I solve this first order second degree differential equation?

    How to solve this first order second degree differential equation ? ##\left(\frac {dy} {dx}\right)^2 + 2x^3 \frac {dy} {dx} - 4x^2y=0 ## Thanks.
  12. Safinaz

    I How to solve this second order differential equation

    Any idea how to solve this equation: ## \ddot \sigma - p e^\sigma - q e^{2\sigma} =0 ## Or ## \frac{d^2 \sigma}{dt^2} - p e^\sigma - q e^{2\sigma} =0 ## Where p and q are constants.Thanks.
  13. B

    How to solve a differential equation for a mass-spring oscillator?

    There is an mass-spring oscillator made of a spring with stiffness k and a block of mass m. The block is affected by a friction given by the equation: $$F_f = -k_f N tanh(\frac{v}{v_c})$$ ##k_f## - friction coefficient N - normal force ##v_c## - velocity tolerance. At the time ##t=0s##...
  14. Meaning

    A Is a solution of a differential equation a function of its parameters?

    Hi everyone, Imagine I have a system of linear differential equations, e.g. the Maxwell equations. Imagine my input variables are the conductivity $\sigma$. Is it correct from the mathematical point of view to say that the electric field solution, $E$, is a function of sigma in general...
  15. potatocake

    How do I solve this differential equation?

    (x cos(y) + x2 +y ) dx = - (x + y2 - (x2)/2 sin y ) dy I integrated both sides 1/2x2cos(y) + 1/3 x3+xy = -xy - 1/3y3+x2cos(y) Then I get x3 + 6xy + y3 = 0 Am I doing the calculations correctly? Do I need to solve it in another way?
  16. Lilian Sa

    First order differential equation involving a square root

    Summary:: solution of first order derivatives we had in the class a first order derivative equation: ##\frac{dR(t)}{dt}=-\sqrt{\frac{2GM(R)}{R}}## in which R dependent of time. and I don't understand why the solution to this equation is...
  17. karush

    MHB Solving a Differential Equation

    $\tiny{1.5.7.19}$ \nmh{157} Solve the initial value problem $y'+5y=0\quad y(0)=2$ $u(x)=exp(5)=e^{5t+c_1}$? so tried $\dfrac{1}{y}y'=-5$ $ln(y)=-5t+c_1$ apply initial values $ln(y)=-5t+ln(2)\implies ln\dfrac{y}{2}=-5t \implies \dfrac{y}{2}=e^{-5t} \implies y=2e^{-5t}$
  18. patric44

    Solution of a parametric differential equation

    hi guys i was trying to solve this differential equation ##\frac{d^{2}y}{dt^{2}}=-a-k*(\frac{dy}{dt})^{3}## in which it describe the motion of a vertical projectile in a cubic resisting medium , i know that this equation is separable in ##\dot{y}## but in order to solve for ##y## it becomes...
  19. F

    Current through Ballistic 2DEG Channel

    So I am a bit uncertain what approach is best for solving this problem and how exactly I should approach it, but my strategy right now is: 1. Solve the time-independent Schrödinger Equation with the given Hamiltonian and find energy eigenvalues of system: -Here I struggle a bit with actually...
  20. AHSAN MUJTABA

    Solving Partial differential equation

    I have tried to do it in standard way by integrating in PDE's but it turned out that ##\psi## is a function of y, so now I have no clue to start this. I know the range of ##\sqrt {g}y## from ##\frac{-\pi}{2}## to ##\frac{\pi}{2}##
  21. P

    Solving a Vector Triangle Differential Equation

    By considering a vector triangle at any point on its circular path, at angle theta from the x -axis, We can obtain that: (rw)^2 + (kV)^2 - 2(rw)(kV)cos(90 + theta) = V^2 This can be rearranged to get: (r thetadot)^2 + (kV)^2 + 2 (r* thetadot)(kV)sin theta = V^2. I know that I must somehow...
  22. chwala

    Solve the Bernoulli differential equation

    kindly note that my question or rather my only interest on this equation is how we arrive at the equation, ##v(x)=ce^{15x} - \frac {3}{17} e^{-2x}## ...is there a mistake on the textbook here? in my working i am finding, ##v(x)=-1.5e^{13x} +ke^{15x}##
  23. chwala

    Is My Solution to the Exact Differential Equation Correct?

    now my approach is different, i just want to check that my understanding on this is correct. see my working below;
  24. PainterGuy

    I Why is this differential equation non-linear?

    Hi, Could you please have a look on the attachment? Question 1: Why is this differential equation non-linear? Is it u=\overset{\cdot }{m} which makes it non-linear? I think one can consider x_{3} , k, and g to be constants. If it is really u=\overset{\cdot }{m} which makes it non-linear then...
  25. chwala

    Solve PDE Uxx+Uyy=-2 with Boundary Conditions

    ial this is the question.
  26. yucheng

    Simmons 7.10 & 7.11: Find Curves Intersecting at Angle pi/4

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  27. Hamiltonian

    I Solving and manipulating the damped oscillator differential equation

    the differential equation that describes a damped Harmonic oscillator is: $$\ddot x + 2\gamma \dot x + {\omega}^2x = 0$$ where ##\gamma## and ##\omega## are constants. we can solve this homogeneous linear differential equation by guessing ##x(t) = Ae^{\alpha t}## from which we get the condition...
  28. Celso

    Superposition in separation method of variables

    Each different boundary condition means a different charge configuration, how can this problem be solved using superposition?
  29. K

    Help solving this Heat Equation please

    I want to solve the heat equation below: I don't understand where the expression for ##2/R\cdot\int_0^R q\cdot sin(k_nr)\cdot r \, dr## came from. The r dependent function is calculated as ##sin(k_nr)/r## not ##sin(k_nr)\cdot r##. I don't even know if ##sin(k_nr)/r## are orthogonal for...
  30. Isaac0427

    I An interesting Nonlinear Differential Equation

    That's pretty much it. If there is a very basic strategy that I am forgetting from ODEs, please let me know, though I don't recall any strategies for nonlinear second order equations. I've tried looking up "motion of a free falling object" with various specifications to try to get the solution...
  31. P

    Trying to solve a transcendental differential equation

    Well, I followed the strategy used by A.S. Parnovsky in his article (\url{http://info.ifpan.edu.pl/firststep/aw-works/fsV/parnovsky/parnovsky.pdf}) and found this differential equation: $$-\frac{g x}{C^{2}} = -\frac{\beta^{2} {y^{\prime}}^{2} \arctan\left({y^{\prime}}\right) + \beta...
  32. diazdaiz

    Find the function of velocity of a person that accelerates to a constant v

    for example, I want to know velocity of a person when time is equal to t, that person start running from 0m/s (t=0s) to max velocity of 1m/s (t=1s). I am thinking that this is like rain droplet that affected by gravity and drag force, where force is directly proportional to its velocity, to make...
  33. docnet

    Solving a second-order differential equation

    Hi all, if anyone could help me solve this 2nd order differential equation, it would mean a lot. Problem: Solve the equation with y = 1, y' = 0 at t = 0 y'' - ((y')^2)/y + (2(y')^2)/y^2 - ((y')^4)/y^4 = 0 I have never solved an ODE of this kind before and I am not sure where to start...
  34. S

    Engineering Write the differential equation that's equivalent to this transfer function

    I have the solution to the problem, and I mechanically, but not theoretically (basically, why do the C(s) and R(s) disappear?), understand how we go from ##(s^5 + 3s^4 + 2s^3 + 4s^2 + 5s + 2) C(s) = (s^4 + 2s^3 + 5s^2 + s + 1) R(s)## to ##c^{(5)}(t) + 3c^{(4)}(t) + 2c^{(3)}(t) + 4c^{(2)}(t) +...
  35. R

    I Is it possible to solve such a differential equation?

    Hello, I would like to is it possible to solve such a differential equation (I would like to know the z(x) function): \displaystyle{ \frac{z}{z+dz}= \frac{(x+dx)d(x+dx)}{xdx}} I separated variables z,x to integrate it some way. Then I would get this z(x) function. My idea is to find such...
  36. S

    MHB Second-Order Nonlinear Differential Equation

    Hi there can someone please help me with this differential equation, I'm having trouble solving it \begin{cases} y''(t)=-\frac{y(t)}{||y(t)||^3} \ , \forall t >0 \\ y(0)= \Big(\begin{matrix} 1\\0\end{matrix} \Big) \ \text{and} \ y'(0)= \Big(\begin{matrix} 0\\1\end{matrix} \Big)\end{cases} \\...
  37. derya

    A Generic Solution of a Coupled System of 2nd Order PDEs

    Hi! I am looking into a mechanical problem which reduces to the set of PDE's below. I would be very happy if you could help me with it. I have the following set of second order PDE's that I want to solve. I want to solve for the generic solutions of the functions u(x,y) and v(x,y). A, B and C...
  38. Butterfly41398

    Can you help me evaluate the integral in this linear differential equation?

    I tried it but I don't know how to evaluate the integral on the last equation. Help.
  39. Vossi

    Electric Circuit Differential equation help

    I'm confused at the part how 4Vc and 48 cos(60t) are deduced, that's all.
  40. karush

    MHB General Solution of Differential Equation System

    $\tiny{27.1}$ 623 Find a general solution to the system of differential equations $\begin{array}{llrr}\displaystyle \textit{given} &y'_1=\ \ y_1+2y_2\\ &y'_2=3y_1+2y_2\\ \textit{solving } &A=\begin{pmatrix}1 &2\\3 &2\end{pmatrix}\\...
  41. S

    Classical mechanics -- Equations for simulating the motion of a body

    Hello forum, i want to make a samulation of a body. The body will be moved horisontal on y,x axis. I want on my simulation the body to change direction many times(for example i want to go for 10sec right and then left end right...). My question is does i need more than one differential equation...
  42. chwala

    Solving this differential equation

    ##-\frac {dy}{dx}=\frac {3+4v}{4-3v}## ##\frac {3+4v}{4-3v}=-v-x\frac {dv}{dx}## ##-\frac {dx}{x}=\frac {4-3v}{8v-3v^2+3}## ##\frac {dx}{x}=\frac {4-3v}{3v^2-8v-3}##=[A/3v+1]+[B/v-3]##
  43. M

    MHB Natural and forced response of a differential equation

    Greetings everyone, I am a bit new to differential equations and I am trying to solve for the natural and forced response of this equation: dx/dt+4x=2sin(3t) ; x(0)=0 Now I know that for the natural response I set the right side of the equation equal to 0, so I get dx/dt+4x=0, thus the...
  44. J

    A Nonlinear Wave Equation (Nonlinear Helmholtz)

    I am trying to solve a PDE (which I believe can be approximated as an ODE). I have tried to solve it using 4th Order Runge-Kutta in MATLAB, but have struggled with convergence, even at an extremely high number of steps (N=100,000,000). The PDE is: \frac{\partial^2 E(z)}{\partial z^2} +...
  45. O

    Problem in solving differential equation

    Hello everyone! I was studying chaotic systems and therefore made some computer simulations in python. I simulated the driven damped anhatmonic oscillator. The problem I am facing is with solving the differential equation for t=0s-200s. I used numpy.linspace(0,200,timesteps) for generate a time...
  46. Amit1011

    What substitution/other method to use to solve this differential equation?

    Differentiating eq1 mentioned above, and using eq 2, i got : $$v\frac{dv}{d\theta}=R\frac{dv}{dt}$$ From this, i got:$$ \frac{d\theta}{dt}=\sqrt{(2/R)(g(1-cos\theta )+asin\theta)}$$ After this point, I am not able to understand what substitution or may be other method could be used to solve...
  47. Ron Burgundypants

    Second order differential equation solution

    I know the solution to the equation (1) below can be written in terms of exponential functions or sin and cos as in (2). But I can't remember exactly how to get there using separation of variables. If I separate the quotient on the left and bring a Psi across, aka separation of variables (as I...
  48. Linder88

    Second order differential equation

    We choose an approximative solution given by $$ u_N(x) = \frac{a_0}{2} + \sum_{n=1}^N a_n \cos nx + b_n \sin nx $$ Comparing this approximative solution with the differential equation yields that $$ \frac{a_0}{2} = a $$ and the boundary conditions yields the equation system $$ a + \sum_{n=1}^N...
  49. M

    A Solving a differential equation with two unknowns

    One thing that is given in paper (attached) is a operating set point for temperature which is given as 20 for day and 16 for night but I do not know whether its initial condition for temperature or not. Can anyone please guide me that what kind of equation is it and how can I solve it with these...
  50. agnimusayoti

    Differential equation on relativistic momentum (ML Boas)

    $$p=\gamma m v$$ $$F = \frac {md (\gamma v}{dt}$$ $$\int{F dt} = \int{md (\gamma v}$$ $$F t= \gamma mv$$ At this step, I don't know how to make v as explicit function of t, since gamma is a function of v too. Thankss
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