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. J

    Determine coefficients of a differential equation

    Hi there. I've been struggling with this problem for days now (4 days, no joke) and I feel like I have a mental block and really cannot get any further. I have a system that's described by f(t) = g''(t) + 15g'(t) + 1600g(t) Where the input is g(t) The problem is to, with this information...
  2. C

    Pressure Testing (furnace installation leak)

    Chasing a ghost perhaps but in the process of pressure testing a simple propane furnace installation I installed a water Column pressure gauge. Over a period of 11 hours the pressure gauge reading will go from 10" WC to 0" WC. In the course of trying to understand this I installed a manometer...
  3. E

    Solve coupled nonlinear differential equations

    Good evening I have these coupled equations and was wondering if there is any chance solving them analytically. If not, how would you approach it numerically? (shown in attachment) Thank you very much
  4. R

    Fin/Extended surface differential equation for temperature

    I'm trying to deduce the differential equation for temperature for a triangular fin: I know that for a rectangular fin, such as: I can do: Energy entering the left: q_x= -kA\frac{dT(x)}{dx} Energy leaving the right: q_{x+dx} = -kA\frac{dT(x)}{dx} - kA\frac{d² T(x)}{dx²}dx Energy lost by...
  5. Tzabcan

    What did I do wrong in solving this basic differential equation?

    I have this basic differential equation du/dt=(u^2)*(sin t) This is obviously a separable diff eq. So what I've done is: g(t) = sin t h(u) = u^2 1/(u^2) du = sin t dt Integrating both side... 1/y = - cos t + c therefor y = - 1/(cos t + c)Which is wrong, there isn't supposed...
  6. Hutchyy

    Differential equation: Autonomous equation question

    Homework Statement I'm stuck on Question #2 part A/B Homework Equations y'=r(1-y/k)y-h=y^2-ky+kh/r y''=2y-k Roots for y'= (k+/-sqrt(k^2-4kh/r))/2 I am assuming the positive root is y2 h<rk/4 [/B] The Attempt at a Solution on part a I'm getting the roots to be y2=(K+sqrt(k^2-4kh/r))/2 and...
  7. S

    Differential Geometry page 193 Nakahara

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  8. Linder88

    Ordinary differential equation with boundary value condition

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  9. H

    Power series solution, differential equation question

    I can not find a solid explanation on this anywhere, so forgive me if this has been addressed already. Given something like y''+y'-(x^2)y=1 or y''+2xy'-y=x, how do I approach solving a differential with a power series solution when the differential does not equal zero? Would I solve the left...
  10. D

    Dupin indicatrix differential geometry

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  11. D

    Unclear concept on simple 2nd order differential eq

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  12. P

    Solve a 2nd order Ordinary Differential Equation

    Homework Statement Y''-((Y')^2)+(C1*exp(Y))=C2 C1 and C2 are constants. exp = e Homework Equations No clue how to start this The Attempt at a Solution Y'=A=dY/dt Y=At+C3 (not sure) A'-(A^2)+C1exp(At+C3)-C2=0 A'-(A^2)+C1exp(C3)exp(At)=0 let C=C1*exp(C3) A'-(A^2)+Cexp(At)=0
  13. T

    MHB Solving 2nd order differential equation numerically

    I'm writing a paper about the projectile motion with the consideration og air resistance - I have obtained two formulas: ax = k*(vx2+vy2)0.5 * vx ay = k*(vx2+vy2)0.5 * vy - g (K and g are constants; K = -0,02, g =9,82) I cand write these two as 2 different differential equations: v'x(t) =...
  14. Z

    Approximate solution of differential equation

    Differential equation: F(y'',y',y,x)=0, y=y(x). Now, there is g=g(x) with F(g'',g',g,x)=δ, where δ is small. Then, can g(x) be taken as an approximate solution of F(y'',y',y,x)=0?
  15. T

    Linear Differential Equation - Initial Value Problem

    Hello, I'm struggling with a simple problem here. It asks me to solve the following initial value problem: So far I've calculated the integration factor μ(x) = ex-x2 and I multiplied both sides of the equation by it and got this...
  16. D

    Motion described by differential equations

    Homework Statement Text from a classical mechanics textbook ( uploaded picture ) shows 2 diff equation describing the motion graphically presented in the uploaded picture. How were these set up? Homework EquationsThe Attempt at a Solution I don't have a slightest clue as how are these...
  17. A

    Can anyone make a differential equation from the below eg:

    [ NOTE ] Thread moved to homework forums by mentor suppose in a dark room a candle is burning, so darkness increases as we move away from the candle. from the below diagram can anyone derive a differential example to show the rate of change of darkness from candle to point B. supposing...
  18. M

    MHB Linear differential equations with constants coefficients

    Hey! :o Each element of the ring $\mathbb{C}[z, e^{\lambda z} \mid \lambda \in \mathbb{C}]$ is of the form $\displaystyle{\sum_{k=1}^n α_kz^{d_k}e^{β_kz}}$. A differential equation in this ring is of the form $$Ly = \sum_{k=0}^m \alpha_k y^{(k)}(z)=\sum_{l=1}^n C_lz^{d_l} e^{\beta_l z} , \ \...
  19. Remixex

    How Do I Solve a Second Order ODE with Non-Constant Coefficients?

    Homework Statement OK, this differential equation was technically created by me, because i need to clear my doubts. Y'' + sqrt(X)*Y' + X^3*Y=3sin(x) and actually just any initial conditions as long as the solution is something i can understand, let me expand my doubt further. I've never solved...
  20. M

    MHB Solutions of a linear differential equation of second order

    Hey! :o I want to check if a linear differential equation of second order has a solution in the ring $\text{Exp}(\mathbb{C})$. We define $\text{Exp}(\mathbb{C})$ as the set of expresions $$\alpha=\alpha_1 e^{\mu_i x}+\dots \alpha_N e^{\mu_N x}$$ where $\alpha_i \in \mathbb{C}$ and $\mu_i \in...
  21. M

    MHB Non homogeneous differential equation - Particular solution

    Hey! :o When we have the non-homogeneous differential equation $$ay''(x)+by'(x)+cy(x)=f(x)$$ and the non-homogeneous term $f(x)$ is of the form $e^{mx}P_n(x)$ we know that the particular solution is $$y_p=x^k(A_0+A_1x+ \dots +A_nx^n)e^{mx}$$ where $k$ is the multiplicity of the eigenvalue...
  22. A

    MHB How to Integrate and Compare Solutions for a Partial Differential System?

    \begin{array}{l} u = u(x,y) \\ v = v(x,y) \\ and\\ {u_x} + 4{v_y} = 0 \\ {v_x} + 9{u_y} = 0 \\ with\ the\ initial\ conditions \\ u(x,0) = 2x _(3)\\ v(x,0) = 3x _(4)\\ \end{array} Easy, u_{xx}-36u_{yy}=0 and v_{xx}-36v_{yy}=0 General solution u\left ( x,y \right )=h\left ( x+6y \right...
  23. L

    SHM - as two ordinary linear differential equations

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  24. Hutchyy

    Ordinary Differential equations question

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  25. M

    MHB Linear differential equation of first order in EXP(C)

    Hey! :o I want to check if there is a solution of a linear differential equation of first order in the ring of exponential sums $\text{EXP}(\mathbb{C})$. I have done the following: The general linear differential equation of first order is $$ax'(z)+bx(z)=y(z) \tag{*}$$ where $x,y \in...
  26. C

    When Will the Curse of the Medicine Man Wipe Out a Tribe?

    Homework Statement This is a interesting (morbid) problem from Simmons- Calculus with Analytic Geometry. In a certain barbourous land, two neighbouring tribes have hated one another from time immemorial. Being barbourous peoples, their powers of belief are strong, and a solemn curse pronounced...
  27. J

    Modeling a Damped Spring System with Differential Equations

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  28. PhotonSSBM

    Application or Theoretical Differential Equations?

    There are two courses I can take for a Differential Equations class at my school. One is for Engineering students and is described this way (I'm a physics major fyi): This course presents an introduction to the theory of differential equations from an applied perspective. Topics include linear...
  29. B

    Charging and discharging capacitor, differential equation

    Homework Statement A circuit consists of a voltage source, voltage ##V## , a resistor, resistance ##R##, and a capacitor, capacitance ##C##, in series. (i) Show that the charge ##Q(t)## in the capacitor satisfies the equation ##R Q' (t) + Q(t)/C = V ##. (ii) Suppose that ##R##, ##C## and...
  30. S

    Solving a first order differential equation

    Homework Statement We have the equation ## (\frac{dr}{ds})^2+(\frac{l}{r})^2=1 ## and want to solve to get ## r=\sqrt{l^2+(s-s_0)^2}## Homework EquationsThe Attempt at a Solution I have worked backwards, plugging in the solution to prove that it is correct, but the closest I have gotten to...
  31. A

    A linear differential equation problem

    Homework Statement A uniform 10-foot-long heavy rope is coiled loosely on the ground. One end of the rope is pulled vertically upward by means of a constant force of 5lb. The rope weighs 1lb/ft. Use Newton's second law to determine a differential equation for the height x(t) of the end above...
  32. M

    But then again, I could be wrong.

    Homework Statement Chemical reactions being studied in which a body A undergoes transformations according to the following scheme: http://prntscr.com/8shuvb k1, k2, k3 , k4 are the rate constants . We denote x (t ), y ( t) , z (t) the respective concentrations of the products A, B, C at a...
  33. B

    Differential Equation problem setup

    Homework Statement Water with a small salt content (5 lb in 1000 gal) is flowing into a very salty lake at the rate of 4 · 105 gal per hr. The salty water is flowing out at the rate of 105 gal per hr. If at some time (say t = 0) the volume of the lake is 109 gal, and its salt content is 107...
  34. ecoo

    Integrating differential equations that have ln

    Hey guys, I have a question concerning the rewriting of a differential equation solution. In the example above, they rewrite [y=(plus/minus)e^c*sqrt(x^2+4)] as [y=C*sqrt(x^2+4)]. I understand that the general solution we get as a result represents all the possible functions, but if we were to...
  35. B

    Differential Equation - Disappearing Term

    Hi, I am struggling to find the solution to the following equation. I can't account for the exponential term, so clearly something is going wrong... 1. Homework Statement Find the general solution to ##x' = tx + 6te^{-t^2}## where ##x(t)##. Homework EquationsThe Attempt at a Solution [/B]...
  36. M

    MHB Existence of solution of a linear differential equation

    Hey! :o I want to check if we can always find a solution of a linear differential equation of first order in the polynomial ring $F[z]$. I have done the following: The general linear differential equation of first order is $$ax'(z)+bx(z)=y(z)$$ where $x,y \in F[z]$. Or is it possible that...
  37. A

    Differential equation for grav, boyant and drag force

    Homework Statement So there is a falling object, you have to take into account the boyant force, the pull of gravity and the drag force A time dependent distance equation is what we're looking for Homework Equations Fd=CdApav2/2 Where Fd is the drag force Cd is the drag coefficient A is the...
  38. D

    Differential equation---a conceptual problem

    Homework Statement I am reading a note on differential equation.There is a point that I don't understand,hopefully someone can explain (Please see the attched) Homework EquationsThe Attempt at a Solution The notes wrote " a1t + b1 x + c1 = a1T + b1X a2t + b2t + c2...
  39. Julio1

    MHB Partial Differential Equations

    Let the PDE $u_{xx}-4u_{xy}+4u_{yy}=0.$ Reduce to the canonical form.Good Morning MHB :). My problem is find the canonical form of the PDE know an variable change. But how I can transform the equation? Thanks.
  40. H

    MHB Please help me solve this differential equation for the initial condition (0,-1)

    Please help me solve this differential equation for the initial condition (0,-1): dx/dy = ((1+x^2)^(1/2))/(xy^3) I think I'm doing something wrong because I end up with ((x^2)(y^3))/2 = ((x^2)+y)^(1/2) + c, but when plugging in the initial condition it ends up being the square root of...
  41. D

    Solve the following differential equation (by regrouping terms)

    Homework Statement (4x^3y^3-2xy)dx+(3x^4y^2-x^2)dy=0[/B]Homework Equations (4x^3y^3-2xy)dx+(3x^4y^2-x^2)dy=0 The Attempt at a Solution i expanded it as 4x^3y^3dx-2xydx+3x^4y^2dy-x^2dy=0 next we have to take the common such that there will be m(X)(xdy+ydx)[/B] m(x) is the common onee
  42. D

    MHB Non exact differential equation

    $y(x^3e^{xy}-y) \, dx+x(xy+x^3e^{xy}) \, dy=0$ change it into exact differential and help me in solving it
  43. W

    Differential Linear Operator Problem not making sense

    Homework Statement I think there may be something wrong with a problem I'm doing for homework. The problem is: Solve the IVP with the differential operator method: [D^2 + 5D + 6D], y(0) = 2, y'(0) = \beta > 0 a) Determine the coordinates (t_m,y_m) of the maximum point of the solution as a...
  44. W

    What math to learn after differential eq. and linear algebra?

    Hi, I'm currently studying to become a chemical engineer. After learning differential equation and linear algebra, I've realized how useful they are in my engineering courses since they make setting up equations and solving them so much easier. So I was wondering if there are other math that...
  45. W

    Differential equation question (I think)

    1. Homework Statement It's to do with mirages, but I don't think the physics context is too important. It's also possible that the solution doesn't involve differential equations, and my method is completely wrong. I've been given that: ##A = \frac{n(1+ay)}{ sqrt(1+(y')^2)}## where y' is...
  46. Y

    Solving an RC circuit using explicit euler

    Homework Statement Hi there. I have a simple RC circuit with a battery voltage of 10 V, R = 1 Ω, C = 1 F and a switch. I want to use the Explicit Euler (forward divided difference) to solve the equation and check for stability, rather than using a ODE. I am finding the equation for when the...
  47. M

    Higher Order Linear Homogenous Differential Equation

    Homework Statement Hi, basically I have a boundary value problem and just want to check that my general solution is correct. x'''' + 16x = 0 Homework EquationsThe Attempt at a Solution I'm pretty sure you make a characteristic equation which would be m4 + 16 = 0. Solving this I get m to be...
  48. Vinay080

    Who invented differential calculus for rational functions?

    Euler mentions in his preface of the book "Foundations of Differential Calculus" (Translated version of Blanton): I don't understand here, who/who all had invented/discovered the study-of-ultimate ratio (differential calculus) for rational functions long before (Newton and Leibniz), without...
  49. Y

    Solve Differential Equation with Euler's Method

    Hi evry body i would like to have an help to resolve this exercice below the followin differential equation with its initial condition dy/dt=-lambda t y(t) t>=0 avec y(0)=y0 where lambda is damping coeficient strictly positive. -find the solution of this equation with Euler's explicite and...
  50. Mark Brewer

    Exact Differential Equations for Level Curves of u(x,y) = cos(x^2-y^2)

    Homework Statement Given u(x.y), find the exact differential equation du = 0. What sort of curves are the solution curves u(x,y) = constant? (These are called the level curves of u). u = cos(x2 - y2)The Attempt at a Solution partial derivative du/dx = (-2x)sin(x2 - y2)dx partial derivative...
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