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. Z

    Difference between resonance in undamped vs damped mass-spring-dashpot

    1) "Undamped system is forced at the same frequency as one of its natural frequencies." Consider the 2nd order differential equation $$\ddot{x}+\omega_0^2x=F_0\cos{\omega t}\tag{1}$$ which models a mass attached to a spring (attached to a wall) with spring constant ##k## and...
  2. Z

    Prove that solutions to autonomous first order DE can't have local maxima

    Here is one argument. Suppose we have a solution ##y## such that ##y(t_0)=y_0## and ##y'(t_0)=f(y_0)=0##. ##y(t)=y_0## is a solution since ##\dot{y}(t)=0## and so ##\dot{y}=f(y)=f(y_0)=0##. I am aware of uniqueness theorems for linear differential equations. I don't remember seeing such a...
  3. M

    Series RLC circuit connected to a DC battery

    How do I solve the differential equation? Please give me a hint.
  4. gigorina

    I LIF Neuron Equation Solution for arbitrary time-dependent current (Neural Dynamics)

    In the book Neural Dynamics: https://neuronaldynamics.epfl.ch/online/Ch1.S3.html There is a solution to the following differential equation (LIF Neuron) for arbitrary time-dependent current. I was trying to figure out the steps the author took to get to the solution. Original Equation: Solution:
  5. G

    Help me solve differential equation please involving a fraction and a square root

    I'm trying to solve the following differential: ##\frac{\dot x}{\sqrt{y(1+\dot x^2)}} = \text{const}## ##\dot x## is the derivative with respect to ##y##. How do I solve it so that I end up with ##x(y)## solution ? You can find this here, but there're 2 problems: 1) I don't understand what...
  6. Graham87

    I Differential equation using power series method

    I am attempting to solve this differential equation with power series I came with the following solution but I doubt it is correct. Since x=1 we get: I doubt its correctness because it looks messy. Also the convergence radian R goes to 0, giving only a solution for x=0 which is not...
  7. T

    I Solutions to Simple Harmonic Motion second order differential equation

    All simple harmonic motion must satisfy $$\frac{d^2s}{dt^2}=-k^2s$$ for a positive value k. The most well known solution is the sinusoidal one $$ s=Acos/sin(\omega t + \delta)$$ A is amplitude, ##\omega##is related to frequency and ##\delta## is phase displacement. My lecturer said that there...
  8. Z

    Apostol Problem on ODE applied to Population Growth

    First of all, a few observations 1) It is not clear if the ##t_1## used in problem 14 is the same ##t_1## from problem 13 where ##x(t_1)=\frac{M}{2}##. However, if it were, then the problem seems like it wouldn't make too much sense because we'd have ##M=2x_1## and that'd be it (though this...
  9. M

    Determine whether ## S[y] ## has a maximum or a minimum

    a) The Euler-Lagrange equation is of the form ## \frac{d}{dx}(\frac{\partial F}{\partial y'})-\frac{\partial F}{\partial y}=0, y(a)=A, y(b)=B ##. Let ## F(x, y, y')=(y'^2+w^2y^2+2y(a \sin(wx)+b \sinh(wx))) ##. Then ## \frac{\partial F}{\partial y'}=2y' ## and ## \frac{\partial F}{\partial...
  10. M

    Proofs about the second-order linear differential equation?

    Proof: (i) Consider the second-order linear differential equation ## \frac{d^2u}{dx^2}+\frac{fu}{2}=0, f=f(x) ##. Then ## u''+\frac{f}{2}u=0\implies r^2+\frac{f}{2}=0 ##, so ## r=\pm \sqrt{\frac{f}{2}}i ##. This implies ## u_{1}=c_{1}cos(\sqrt{\frac{f}{2}}x) ## and ##...
  11. C

    I Question about solving linear first order non-homogeneous ODEs

    A general equation for linear first order non-homogeneous ODE is: ## y' + a(x)y = b(x) ##. The procedure to solve ( assuming ## a(x) , b(x) ## are continuous so that the fundamental theorem of calculus could be used ) it is to multiply it by ## e^{A(x)} ## ( here ## A'(x) = a(x) ## ) s.t. ##...
  12. chwala

    Solve the given differential equation

    My interest is only on the highlighted part, i can clearly see that they made use of chain rule i.e by letting ##u=1+x^2## we shall have ##du=2x dx## from there the integration bit and working to solution is straightforward. I always look at such questions as being 'convenient' questions. Now...
  13. chwala

    Solve the given differential equation

    I am on differential equations today...refreshing. Ok, this is a pretty easier area to me...just wanted to clarify that the constant may be manipulated i.e dependant on approach. Consider, Ok I have, ##\dfrac{dy}{6y^2}= x dx## on integration, ##-\dfrac{1}{6y} + k = \dfrac{x^2}{2}##...
  14. U

    A Solve PDE w/ Comsol 5.3: Numerical Solution & Time Evolution

    What is the best way to solve numerically the following equation using Comsol 5.3. ##\frac{\partial T}{\partial t}=\frac{\partial ^2T}{\partial x^2}+\text{St}\left[1+\left(\frac{\partial T}{\partial x}\right)_{x=0}\right]\frac{\partial T}{\partial x}## ##T(0,t)=1## ##T(\infty ,t)=0##...
  15. L

    Solve these two coupled first-order differential equations and sketch the flow

    Hi, unfortunately, I have a problem to solve the following task The equation looks like this: $$\left(\begin{array}{c} \frac{d}{dt} x(t) \\ \frac{d}{dt} y(t) \end{array}\right)=\left(\begin{array}{c} -a y(t) \\ x(t) \end{array}\right)$$ Since the following is true ##\frac{d}{dt}...
  16. L

    Mathematica Problems solving this differential equation for a Pendulum with Mathematica

    Hi, unfortunately, I have problems that Mathematica does not solve the differential equation. The task is as follows and it is about the task c In the Mathematica Notebook, the following was written for task c "You can use the following two lines of code to produce the solutions of the...
  17. L

    Linear first-order differential equation with an initial condition

    Hi, unfortunately I have problems with the task d and e, the complete task is as follows: I tried to form the derivative of the equation ##f(x)##, but unfortunately I have problems with the second part, which is why I only got the following. $$\frac{d f(x)}{dx}=f_0 g(x) \ exp\biggl(...
  18. C

    Differential equation problem: y" + y' - 2y = x^2

    For this, The solution is, However, why did they not move the x^2 to the left hand side to create the term ##(-2A - 1)x^2##? Is it possible to solve it this way? Many thanks!
  19. pawlo392

    A Differential equation and Appell polynomials

    Hello! Let $n$ be a natural number, $P_n(x)$ be a polynomial with rational coefficients, and $\deg P_n(x) = n$. Let $P_0(x)$ be a constant polynomial that is not equal to zero. We define the sequence ${P_n(x)}_{n \geq 0}$ as an Appell sequence if the following relation holds: \begin{equation}...
  20. gurbir_s

    A Solving this first-order differential equation for neutron abundance

    The time rate of change of neutron abundance ##X_n## is given by $$\frac{dX_n}{dt} = \lambda - (\lambda + \hat\lambda)X_n$$ where ##\lambda## is neutron production rate per proton and ##\hat\lambda## is neutron destruction rate per neutron. Given the values of ##\lambda## and ##\hat\lambda## at...
  21. E

    I Fundamental matrix of a second order 2x2 system of ODEs

    Let ## \mathbf{x''} = A\mathbf{x} ## be a homogenous second order system of linear differential equations where ## A = \begin{bmatrix} a & b\\ c & d \end{bmatrix} ## and ## \mathbf{x} = \begin{bmatrix} x(t)\\ y(t)) \end{bmatrix} ## Now to solve this equation we transform it into a 4x4...
  22. PhysicsRock

    Solution for differential equation

    Greetings, in one of the exercise sheets we were given by our Prof, we were supposed to draw the trajectory of a patricle that moves toward a bounded spherical potential that satisfies ## V(\vec{r}) = \begin{cases} V_0 & | \vec{r} | \leq a \\ 0 & else \\ \end{cases} ## for...
  23. N

    Unclear differential equation from a thermodynamics textbook

    In the thermodynamics textbook there is written: 𝛿𝐴 = 𝑇𝑑𝑆 βˆ’ π‘‘π‘ˆ = 𝑑(𝑇𝑆) βˆ’ 𝑆𝑑𝑇 βˆ’ π‘‘π‘ˆ = βˆ’π‘‘(π‘ˆ βˆ’ 𝑇𝑆) βˆ’ 𝑆𝑑𝑇 = βˆ’π‘‘πΉ βˆ’ 𝑆𝑑𝑇 How did we get the bolded area from TdS? Is that property of derivative, integral, or something else :/
  24. chwala

    Solve the given partial differential equation

    Looking at pde today- your insight is welcome... ##Ξ·=-6x-2y## therefore, ##u(x,y)=f(-6x-2y)## applying the initial condition ##u(0,y)=\sin y##; we shall have ##\sin y = u(0,y)=f(-2y)## ##f(z)=\sin \left[\dfrac{-z}{2}\right]## ##u(x,y)=\sin \left[\dfrac{6x+2y}{2}\right]##
  25. chwala

    Solve the given first order differential equation

    My thinking is two-fold, firstly, i noted that we can use separation of variables; i.e ##\dfrac{dy}{y}= \sec^2 x dx## on integrating both sides we have; ##\ln y = \tan x + k## ##y=e^{\tan x+k} ## now i got stuck here as we cannot apply the initial condition ##y(\dfrac {Ο€}{4})=-1##...
  26. C

    Solving the SHM differential equation

    I am trying to solve this homogenous linear differential equation . Since it is linear, I can use the substitution . Which gives, (line 1) (line 2) (line 3) (line 4) (line 5) Which according to Morin's equals, (line 6) However, could someone please show me steps how he got from line 5 to 6...
  27. C

    Morin classical mechanics differential equation problem

    I was reading the oscillations chapter which was talking about how to solve linear differential equations. He was talking about how to solve the second order differential below, where a is a constant: In the textbook, he solved it using the method of substitution i.e guessing the solution...
  28. chwala

    A Solve the Partial differential equation ##U_{xy}=0##

    This is part of the notes; My own way of thought; Given; ##U_{xy}=0## then considering ##U_x## as on ode in the ##y## variable; we integrate both sides with respect to ##y## i.e ##\dfrac{du}{dx} \int \dfrac{1}{dy} dy=\int 0 dy## this is the part i need insight...the original problem...
  29. Shovon00000

    I Expressing a differential equation into a different format

    How do we express this differential equation (dy/dx)= (y/x) + tan(y/x) into this form( Mdx + Ndy=0) where M,N are functions of (x,y) ?
  30. H

    I Second order non-homogeneous linear ordinary differential equation

    I shall not begin with expressing my annoyance at the perfect equality between the number of people studying ODE and the numbers of ways of solving the Second Order Non-homogeneous Linear Ordinary Differential Equation (I'm a little doubtful about the correct syntactical position of 'linear')...
  31. G

    I Is there a name for this sort of differential equation?

    Is there a name to this sort of differential equation? $$ f(z) + 2zf'(z) + f''(z) = 0 ~. $$ I ran into it somewhere and it does not look to be Hermite. I think it has the general solution $$ f(z) = e^{-z^2} \big( c_1 + c_2 \Phi(\sqrt{3}z) \big) \quad \textnormal{($\Phi(x)$ is probit function.)}...
  32. H

    Repeated roots of a characteristic equation of third order ODE

    The characteristic equation ## m^3 -6m^2 + 12m -8 = 0## has just one single, I mean all three are equal, root ##m=2##. So, one of the particular solution is ##y_1 = e^{2x}##. How can we find the other two? The technique ##y_2 = u(x) e^{2x}## doesn't seem to work, and even if it were to work how...
  33. S

    Differential equation of vector field

    I was thinking of using the chain rule with dF/dx = 0i + (3xsin(3x) - cos(3x))j and dF/dy = 0i + 0j but dF/dy is still a vector so how can it be inverted to get dy/dF ? what are the other methods to calculate this?
  34. A

    Solution to Differential Equation with Limit Boundary Condition

    The original differential equation is: My solution is below, where C and D are constants. I have verified that it satisfies the original DE. When I apply the first boundary condition, I obtain that , but I'm unsure where to go from there to apply the second boundary condition. I know that I...
  35. N

    What does the differential equation answer mean?

    Hi, last semester I "solved" a full differential equation and the answer was (see the picture). What does it mean? Can I make a graphic with it or what? I really don't get it. *Arrows are just a continuation of the main formula*
  36. dim_d00m

    A Recurrence relations for series solution of differential equation

    I am currently looking at section IIA of the following paper: https://arxiv.org/pdf/gr-qc/0511111.pdf. Eq. (2.5) proposes an ansatz to solve the spheroidal wave equation (2.1). This equation is $$ \dfrac{d}{dx} \left((1-x^2) \dfrac{d}{dx}S_{lm} \right) + \left(c^2x^2 + A_{lm} -...
  37. C

    A FEM basis polynomial order and the differential equation order

    Is there a good rubric on how to choose the order of polynomial basis in an Finite element method, let's say generic FEM, and the order of the differential equation? For example, I have the following equation to be solved ## \frac{\partial }{\partial x} \left ( \epsilon \frac{\partial u_{x}...
  38. A

    Engineering How do I find the differential equation for this circuit?

    Hi, The following circuit is given, where the switch S is closed at time t=0. a) Set up the general differential equation (DE) for the current i(t) and bring the result into the following form## \frac{di(t)}{dt} +c_1 i(t)=c_0,## with the constant terms c0 and c1. Hint: Determine the DE using...
  39. chwala

    Solve the problem involving differential equation

    My approach for part (a), ##\dfrac{dp}{dt}+P(t)=100## I.f=## e^{\int 1 dt} = e^t## Therefore, ##(e^t p)^{'}=100e^t## ##e^tp=\int 100e^t dt## ##e^tp=100e^t+k## Applying initial condition, ##p(0)=2000## ##2000=100+k## ##k=1900## Therefore, ##e^tp=100e^t+1900## ##p=100+1900e^{-t}##...
  40. MatinSAR

    Find the differential equation

    Hi ... I have written the equation of family of straight lines which are tangent to the circle as : y=(-m/n)x+(m^2/n)+n line intersects circle at : (m,n) But I can't understand how to find differential equation of this ... I will be appreciated if anyone has extra time to give me a little...
  41. A

    How did they get to this differential equation?

    Hello! Disclaimer: I am not really sure in which forum I should post this problem since the homework is electrical engineering,but the problem I am facing is of mathematical nature (at least I think). Consider this circuit; The given RC network contains the resistors R1 = 200 Ξ© and R2 = 300...
  42. Haorong Wu

    Solving a partial differential equation

    If the right-hand side is zero, then it will be a wave equation, which can be easily solved. The right-hand side term looks like a forced-oscillation term. However, I only know how to solve a forced oscillation system in one dimension. I do not know how to tackle it in two dimensions. I have...
  43. D

    Finding the differential equation of motion

    Summary:: Differential equation of motion, parabola Hi. I've tried resolve this problem but I have two doubts. The first is about the differential equation of motion because I can't simplify it to the form y" + a*y' + b*y = F(t). I'm not sure if what I got is right. My second doubt is that I...
  44. S

    Classification of a second order partial differential equation

    Hello! Consider this partial differential equation $$ zu_{xx}+x^2u_{yy}+zu_{zz}+2(y-z)u_{xz}+y^3u_x-sin(xyz)u=0 $$ Now I've got the solution and I have a few questions regarding how we get there. Now we've always done it like this.We built the matrix and then find the eigenvalues. And here is...
  45. V

    I Limit cycles, differential equations and Bendixson's criterion

    I have a few questions about the negative Bendixon criterion. In order to present my doubts, I organize this post as follows. First, I present the theorem and its interpretation. Second, I present a worked example and my doubts. The Bendixson criterion is a theorem that permits one to establish...
  46. J

    I Integral-form change of variable in differential equation

    I have the following differential equation, which is the general Sturm-Liouville problem, $$ \dfrac{d}{dx} \left[ p(x) \dfrac{d\varphi}{dx} \right] + \left[ \lambda w(x) - q(x) \right] \varphi(x) = 0\ , $$ and I want to perform the change of variable $$ x \rightarrow y = \int_a^x \sqrt{\lambda...
  47. chwala

    Solve the first order differential equation

    From my working...I am getting, ##xy=####\int x^{-1/2}\ dx## ##y##=##\dfrac {2}{x}##+##\dfrac {k}{x}## ##y##=##\dfrac {2}{x}##+##\dfrac {6}{x}## ##y##=##\dfrac {8}{x}## i hope am getting it right...
  48. L

    I Problem with integrating the differential equation more than once

    Starting from equation \frac{dy}{dx}=\int^x_0 \varphi(t)dt we can write dy=dx\int^x_0 \varphi(t)dt Now I can integrate it \int^{y(x)}_{y(0)}dt=\int^x_0dx'\int^x_0\varphi(t)dt Is this correct? Or I should write it as \int^{y(x)}_{y(0)}dt=\int^x_0dx'\int^{x'}_0\varphi(t)dt Best wishes in new year...
  49. chwala

    Solve this first order differential equation

    This is the question; This is the solution; Find my approach here, ##x####\frac {dy}{dx}##=##1-y^2## β†’##\frac {dx}{x}##=##\frac {dy}{1-y^2}## I let ##u=1-y^2## β†’ ##du=-2ydy##, therefore; ##\int ####\frac {dx}{x}##=##\int ####\frac {du}{-2yu}##, we know that ##y##=##\sqrt {1-u}## ##\int...
  50. 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...
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