What is Differential: 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. chwala

    Clarifying the Use of Integrating Factors in Exact Differential Equations

    I am looking at this and i would like some clarity... at the step where "he let" ##μ_y##=0" Could we also use the approach, ##μ_x##=0"? so that we now have, ##μ_y##M=μ(##N_{x} -M_{y})##... and so on, is this also correct?
  2. jk22

    I Why the integral of a differential does not give the function back in 2D?

    Let f be a 2 variables function. 1) ##f(x,y)=g(x)+h(y)\Rightarrow df=g'(x)dx+h'(y)dy\Rightarrow\int df=g(x)+k(y)+h(y)+l(x)=f(x,y),\textrm{ if } k=l=0## 2) ##f(x,y)=xy\Rightarrow df=ydx+xdy\Rightarrow\int df=2xy+k(y)+l(x)\neq f(x,y)## Why in the second case the function cannot be recovered ?
  3. 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...
  4. 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...
  5. guyvsdcsniper

    Courses Where to go after Differential Equations?

    I am currently pursuing a Bachelors in Physics. With my current work experience, that degree will eventually allow me to reach an engineering position in Non Destructive Testing. While I enjoy the career field I believe I could do more with my degree. I personally would like to work at LHC or...
  6. 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.
  7. T

    A Differential Geometry Class: Suggestions Welcome

    Can anyone recommend a good on-line class for differential geometry? I'd like to start studying GR but want a good background in differential geometry before doing so. Many thanks.
  8. Hamiltonian

    Moment of inertia of a disc using rods as differential elements

    I know there are more convenient differential elements that can be chosen to compute the moment of inertia of a disc(like rings). the mass of the differential element: $$dm = (M/\pi R^2) (dA) = (M/ \pi R^2) (2\sqrt{R^2 - y^2})(dy)$$ the moment of inertia of a rod through its COM is...
  9. 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.
  10. AHSAN MUJTABA

    A Phase Portraits of a system of differential equations

    One thing that bothers me regarding the phase portraits, if I plot a phase portrait, then all my possible solutions (for different initial conditions) are included in the diagram? In other words, a phase portrait of a system of ODE's is its characteristic diagram?
  11. Leo Liu

    What does the square of a differential mean?

    When I was following the calculations of finding the potential energy of a spring standing on a table under gravity, I encountered the integral shown below, where ##d\xi## is the compression of a tiny segment of the spring and ##k'## is the effective spring constant of that segment. The integral...
  12. greg_rack

    Learning DEs: Solving 2nd Order Differential Equations

    Hi guys, I have just started studying DEs on my own, so pardonne moi in advance for the probably silly question :) Via Newton's second law of motion: $$x''=\frac{F}{m} \ [1]$$ Which is a second-order differential equation. But, from here, how do I get the good old equation of motion...
  13. greg_rack

    B Real world applications of differential equations

    Hi guys, how are you doing? My maths teacher asked me to work on and deliver an engaging insight-oriented "lesson" to my class, about physical/engineering and real-world applications of differential equations, in order to better get the meaning of operating with such mathematical objects. Of...
  14. 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##...
  15. E

    Studying Need some advice -- Studying oscillations before differential equations?

    Hello there, I need some advice here. I am currently studying intro physics together with calculus. I am currently on intro to oscillatory motion and waves (physics-wise) and parametric curves (calc/math-wise). I noticed that in the oscillatory motion section, I need differential equation...
  16. 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...
  17. potatocake

    Is My Solution to This Exact Differential Equation Correct?

    (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?
  18. D

    I Commutative algebra and differential geometry

    In Miles Reid's book on commutative algebra, he says that, given a ring of functions on a space X, the space X can be recovered from the maximal or prime ideals of that ring. How does this work?
  19. Falgun

    Applied Ordinary Differential Equations Books

    I am trying to self study Ordinary Differential Equations and am totally fed up of "cookbook style ODEs". I have recently finished Hubbard's Multivariable Calculus Book and Strang's Linear algebra book. I would like a rigorous and Comprehensive book on ODEs. I have shortlisted a few books below...
  20. K

    A Differential forms on R^n vs. on manifold

    First time looking at differential forms. What is the difference of the forms over R^n and on manifolds? Does the exterior product and derivative have different properties? (Is it possible to exaplain this difference without using the tangent space?)
  21. 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...
  22. 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 $ln(y)=-5t+ln(2)\implies ln\dfrac{y}{2}=-5t \implies \dfrac{y}{2}=e^{-5t} \implies y=2e^{-5t}$
  23. 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...
  24. C

    Proving Poincare Algebra Using Differential Expression of Generator

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  25. Differential (steering gear box)

    Differential (steering gear box)

    Around the corner how a differential steering works 1975 ! Engineering Books Join our group : https://www.facebook.com/groups/2373375319542186/
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  26. 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}##
  27. 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...
  28. 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}##
  29. 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;
  30. Leo Liu

    A constrained differential probelm

    Define that $$w(x,y,z)=zxe^y+xe^z+ye^z$$ So the constraint equation is ##x^2y+y^2x=1##. And its differential is ##dy=-\frac{2xy+y^2}{2xy+x^2}##. However, the solution plugs in ##z=0## when computing ##\frac{\partial w}{\partial x}## as shown in the screenshot below. While I understand that...
  31. 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...
  32. chwala

    Solve PDE Uxx+Uyy=-2 with Boundary Conditions

    ial this is the question.
  33. yucheng

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

    >10. Let a family of curves be integral curves of a differential equation ##y^{\prime}=f(x, y) .## Let a second family have the property that at each point ##P=(x, y)## the angle from the curve of the first family through ##P## to the curve of the second family through ##P## is ##\alpha .## Show...
  34. V

    Solve the system of differential equations

    I have my set of differential equations which is dx/dt = -2x, dy/dt=-y+x2, with the initial conditions x(0)=x0 and y(0)=y0. I'm a little confused about how to approach this problem. I thought at first I would differentiate both sides of dx/dt = -2x in order to get d2x/dt2 = -2, and then I would...
  35. 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...
  36. 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...
  37. 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...
  38. E

    Lie derivative of general differential form

    The first two parts I think were fine, I expressed the tensors in coordinate basis and wrote for the first part$$ \begin{align*} \mathcal{L}_X \omega = \mathcal{L}_X(\omega_{\nu} dx^{\nu} ) &= (\mathcal{L}_X \omega_{\nu}) dx^{\nu} + \omega_{\nu} (\mathcal{L}_X dx^{\nu}) \\ &= X^{\sigma}...
  39. T

    A Dx in an integral vs. differential forms

    Good Morning To cut the chase, what is the dx in an integral? I understand that d/dx is an "operator" on a function; and that one should never split, say, df, from dx in df/dx That said, I have seen it in an integral, specifically for calculating work. I do understand the idea of...
  40. JD_PM

    Solving a system of differential equations

    Summary:: We want to find explicit functions ##g(y,t)## and ##f(y,t)## satisfying the following system of differential equations. I attached a very similar solved example. Given the following system of differential equations (assuming ##y \neq 0##) \begin{equation*} -y\partial_t \left(...
  41. 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...
  42. P

    I Solving a system of differential equations by elimination

    I would to know if I'm solving system differential equation by elimination correctly. Could somebody check my sample task and tell if something is wrong?
  43. yucheng

    Applied Is Piaggio's Differential Equations worth reading?

    I got to know of this book through Freeman Dyson's obituary. Just wondering, is it useful in studying Physics (it seems to cover everything), do people even use it these days? I understand differential equations are basically half of Physics. By the way, this book is really old, are there any...
  44. 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) +...
  45. 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...
  46. 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} \\...
  47. J

    I Integration of differential forms

    I am confused as to how exactly we integrate differential forms. I know how to integrate them in the sense that I can perform the computations and I can prove statements, but I don't understand how it makes sense. Let's integrate a 1-form over a curve for example: Let ##M## be a smooth...
  48. J

    Differential Amplifier with Two stages

    What I tried to do : First I tried to calculte vb1 : Saturation => Vds > Vgs - vth Vds > Vgs - vth Vd - Vs > Vgs - vth Vd= Vb1 Vb1 > Vgs - vth + Vs Vs= Vss Result : Vb1 > 0.4 - vth + Vs I don't know if it's correct and don't know what to do for the two others.
  49. J

    I Testing my knowledge of differential forms

    I am test my knowledge of differential forms and obviously I am missing something because I can't figure out where I am going wrong here: Let ##C## denote the positively oriented half-circle of radius ##r## parametrized by ##(x,y) = (r \cos t, r \sin t)## for ##t \in (0, \pi)##. The value of...
  50. 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.
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