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. 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
  2. S

    Solve this differential equation for the curve & tangent diagram

    Here is my attempt at a solution: y = f(x) yp - ym = dy/dx(xp-xm) ym = 0 yp = dy/dx(xp-xm) xm=ypdy/dx + xm xm is midpoint of OT xm = (ypdy/dx + xm) /2 Not sure where to go from there because the solution from the link uses with the midpoint of the points A and B intersecting the x-axis...
  3. H

    What is the Differential Equation for Airflow in a Balloon?

    I have a problem. The task is to develop an differential equation of the airflow of a balloon. I know that it is dependent on the volume and pressure. But I can't get a good differential equasion. Can someone help me? [Thread moved from the technical forums, so no Homework Help Template is shown]
  4. mertcan

    A Solution to a differential equation with variable coefficients

    Hi, I really struggled to dig valuable things out of internet and books related to high order homogeneous differential equation with variable coefficients but I have nothing. All methods I see involves given solution and try to find others(like reduction of order method), even for second order...
  5. E

    B A differential equation, or an identity?

    This is quite literally a showerthought; a differential equation is a statement that holds for all ##x## within a specified domain, e.g. ##f''(x) + 5f'(x) + 6f(x) = 0##. So why is it called a differential equation, and not a differential identity? Perhaps because it only holds for a specific set...
  6. K

    TISE solution for a hydrogen atom

    I am unable to complete the first part of the question. After I plug in the function for psi into the differential equation I am stuck: $$\frac {d \psi (r)}{dr} = -\frac 1 a_0 \psi (r), \frac d{dr} \biggl(r^2 \frac {d\psi (r)}{dr} \biggr) = -\frac 1 {a_0}\frac d {dr} \bigl[r^2 \psi(r) \bigr] =...
  7. K

    I Differential equation with two terms

    I'm trying to solve a differential equation of the form $$\frac{A'(x)}{A(x)}f(x,y) = \frac{B'(y)}{B(y)}$$ where prime denotes differentiation. I know that for the case ##f(x,y) = \text{constant}## we just equal each side to a same constant. Can I do that also for the case where ##f(x,y)## is not...
  8. S

    I How does one solve Uxx+Uyy+Uzz=C when C is non-zero?

    How does one solve the partial differential equation Uxx+Uyy+Uzz=C when C is non-zero. Here U is a function of x,y and z where (x,y,z) lies in the ball centered at 0 of radius 1 and U=0 on the boundary. Uxx, Uyy and Uzz denote second partial derivatives with respect to x, y and z. Any hints on...
  9. M

    A Partial Differential equation (Heat eqn)

  10. SilverSoldier

    Differential Equation for a Pendulum

    Suppose we displace the pendulum bob ##A## an angle ##\theta_0## initially, and let go. This is equivalent to giving it an initial horizontal displacement of ##X## and an initial vertical displacement of ##Y##. Let ##Y## initially be a negative number, and ##X## initially be positive. I observe...
  11. E

    Solving an exact differential equation

    I let ##M = 4xy + 1## and ##N = 2x^2 + \cos{(y)}##. Since ##\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}##, the equation is exact and we have $$\frac{\partial f(x,y)}{\partial x} = 4xy + 1$$ From inspection, you can tell this has to lead to $$f(x,y) = 2x^2 y + x + h(y)$$ and we...
  12. Tony Hau

    I Deriving the differential equation for the underdamped case

    The formula for general oscillation is: The formula for underdamping oscillation is: where λ = -γ +- sqart(γ^2 - ω^2), whereas A+ and A- , as well as λ+ and λ-, are complex conjugates of each other. After some operations, we get: x(t) = Ae^(-γx)[e^i(θ+ωx) +e^-i(θ+ωx)], where A is the modulus...
  13. caffeinemachine

    MHB Maximum value a function satisfying a differential equation can achieve.

    Let $f:\mathbb R\to \mathbb R$ be a twice-differentiable function such that $f(x)+f^{\prime\prime}(x)=-x|\sin(x)|f'(x)$ for $x\geq 0$. Assume that $f(0)=-3$ and $f'(0)=4$. Then what is the maximum value that $f$ achieves on the positive real line? a) 4 b) 3 c) 5 d) Maximum value does not exist...
  14. C

    A Partial differential equation containing the Inverse Laplacian Operator

    I am trying to reproduce the results of a thesis that is 22 years old and I'm a bit stuck at solving the differential equations. Let's say you have the following equation $$\frac{\partial{\phi}}{\partial{t}}=f(\phi(r))\frac{{\nabla_x}^2{\nabla_y}^2}{{\nabla}^2}g(\phi(r))$$ where ##\phi,g,f## are...
  15. Ayoub Tamin

    Mathematica How to solve this differential equation using Mathematica's Dsolve?

  16. D

    I Help getting started with this differential equation

    I need to solve ∂2Φ/∂s2 + (1/s)*∂Φ/ds - C = 0 Where s is a radial coordinate and C is a constant. I know this is fairly simple but I haven't had to solve a problem like this in a long time. Can someone advise me on how to begin working towards a general solution? Is the method of...
  17. J

    Modeling the populations of foxes and rabbits given a baseline

    From solving the characteristic equations, I got that ##\lambda = 0.5 \pm 1.5i##. Since using either value yields the same answer, let ##\lambda = 0.5 - 1.5i##. Then from solving the system for the eigenvector, I get that the eigenvector is ##{i}\choose{1.5}##. Hence the complex solution is...
  18. karush

    MHB Apc.9.3.1 solution to the differential equation condition

    253 Which of the following is the solution to the differential equation condition $$\dfrac{dy}{dx}=2\sin x$$ with the initial condition $$y(\pi)=1$$ a. $y=2\cos{x}+3$ b. $y=2\cos{x}-1$ c. $y=-2\cos{x}+3$ d. $y=-2\cos{x}+1$ e. $y=-2\cos{x}-1$ integrate $y=\displaystyle\int 2\sin...
  19. T

    Variations of a parameter in a differential equation

    I tried to derive this by myself but I'm stuck. What i did it to substitute a_{1} with a_{1} +\Delta a_{1} in the first equation, getting: (a_{1}+\Delta a_{1})\dot{y}(t)+y(t)=b_{0}u(t)+b_{1}\dot{u}(t) and trying to subtract a_{1}\dot{y}(t)+y(t)=b_{0}u(t)+b_{1}\dot{u}(t) to it. But it's not...
  20. A

    I Constants at the end of the Frobenius method

    I'm having a hard time grasping the concept of reducing the two recursive relations at the end of the frobenius method. For example, 2xy''+y'+y=0 after going through all the math i get y1(x) = C1[1-x+1/6*x^2-1/90*x^3+...] y2(x) = C2x^1/2[1-1/3*x+1/30*x^2-1/630*x^3+...] I know those are right...
  21. chaksome

    Solution for a second-order differential equation

    I wish to know if there is a method to work out x(t). [No matter which form f(t) is] Thank you~
  22. chwala

    Solve the differential equation

    on introducing a term on both sides, we have ##(x^2+xy-2xy)y^{'}=x^2+y^2-2xy## ##(x^2-xy)y^{'}=(x-y)^2## ##x(x-y)y^{'}=(x-y)^2## ##xy^{'}=(x-y)## ##y^{'}=1-y/x## ## v+x v^{'}=1-v## ...ok are the steps correct before i continue?
  23. anooja559

    Solution for a second order differential equation

    Hi, Could you please help me to solve a second-order differential equation given below ∂M/r∂r+∂2M/∂r2 = A [Moderator's note: Moved from a technical forum and thus no template.]
  24. B

    First-order nonlinear differential equation

    Homework Statement: first order non linear equation Homework Equations: dT/dt=a-bT-Z[1/(1+vt)^2]-uT^4 a,b,z,v,u are constant t0=0 , T=T0 Hi, i need find an experession of T as function of t from this first order nonlinear equation: dT/dt=a-bT-Z[1/(1+vt)^2]-uT^4 a,b,z,v,u are constant...
  25. giveortake

    Engineering Dirac Delta Function in an Ordinary Differential Equation

    1.) Laplace transform of differential equation, where L is the Laplace transform of y: s2L - sy(0) - y'(0) + 9L = -3e-πs/2 = s2L - s+ 9L = -3e-πs/2 2.) Solve for L L = (-3e-πs/2 + s) / (s2 + 9) 3.) Solve for y by performing the inverse Laplace on L Decompose L into 2 parts: L =...
  26. K

    Differential equation problem: Modeling the spread of a rumor on campus

    So this is what I have done: ##f'(t)=k*f(t)*(A-f(t))*(1-sin(\frac{pi*x}{12}))## ##\frac{1}{f(t)*(A-f(t))}=k*(1-sin(\frac{pi*x}{12}))## I see that the left can be written as this (using partial fractions): ##1/A(\frac{1}{f(t)}-\frac{1}{A-f(t)})## And then I take the integral on both sides and...
  27. TheMercury79

    I Scale factor from Friedmann's equations

    If we take a flat universe dominated by radiation, the scale factor is ##a(t)=t^{1/2}## which can be derived from the first Friedmann Equation:$$(\dot a/a)^2 = \frac{8\pi G}{3c^2}\varepsilon(t)-\frac{kc^2}{R_0^2 a(t)^2}$$ But suppose I want to show this using the second Friedmann Equation (Also...
  28. Terrycho

    Partial Differential Equation: a question about boundary conditions

    Consider the following linear first-order PDE, Find the solution φ(x,y) by choosing a suitable boundary condition for the case f(x,y)=y and g(x,y)=x. --------------------------------------------------------------------------- The equation above is the PDE I have to solve and I denoted the...
  29. S

    Solving this partial differential equation

    Introducing the new variables ##u## and ##v##, the chain rule gives ##\dfrac{{\partial{f}}}{{\partial{x}}}=\dfrac{{\partial{f}}}{{\partial{u}}} \dfrac{{\partial{u}}}{{\partial{x}}}+\dfrac{{\partial{f}}}{{\partial{v}}} \dfrac{{\partial{v}}}{{\partial{x}}}##...
  30. Sara_76

    B Solve a second-order differential equation

  31. W

    I How can we identify non-linear singular differential equation

    i am doing research to make criteria by which i can identify easily linear and non-linear and also identify its singular or not by doing simple test.please help me in this regard.
  32. Luke Tan

    I Can the ODE \psi''-y^2\psi=0 be solved using a general method?

    When reading through Shankar's Principles of Quantum Mechanics, I came across this ODE \psi''-y^2\psi=0 solved in the limit where y tends to infinity. I have tried separating variables and attempted to use an integrating factor to solve this in the general case before taking the limit, but...
  33. B

    Solve the differential equation: y′′y′+yy′+yy′′=0

    I tried the substitution ##y=e^{\int z(x)}##,##z(x)## is an arbitrary function to be determined. Substitute this to the original differential equation,and dividing ##y^2## yields ##(z+1)z'+z^3+z^2+z=0##,which is a first order differential equation. Trying to solve this first order differential...
  34. J

    MHB Differential equation - distance needed to achieve target speed

  35. S

    Set up the differential equation showing the voltage V(t) for this RC circuit

    Hi all. I have another exam question that I am not so sure about. I've solved similar problems in textbooks but I have a feeling once again that the correct way to solve this problem is much simpler and eluding me. Especially because my answer to a) is already the solution to c) and d) (I did...
  36. R

    I Does this ODE have any real solutions?

    The ODE is: \begin{equation} (y'(x)^2 - z'(x)^2) + 2m^2( y(x)^2 - z(x)^2) = 0 \end{equation} Where y(x) and z(x) are real unknown functions of x, m is a constant. I believe there are complex solutions, as well as the trivial case z(x) = y(x) = 0 , but I cannot find any real solutions. Are...
  37. murshid_islam

    Differential equation problem: Solve dy/dx = (y^2 - 1)/(x^2 - 1), y(2) = 2

    This is my attempt: \frac{dy}{dx} = \frac{y^2 - 1}{x^2 - 1} \\ \int \frac{dy}{y^2 - 1} = \int \frac{dx}{x^2 - 1} \\ \ln \left| \frac{y-1}{y+1} \right| + C_1 = \ln \left| \frac{x-1}{x+1} \right| + C_2 \\ \ln \left| \frac{y-1}{y+1} \right| = \ln \left| \frac{x-1}{x+1} \right| + C Since y(2) =...
  38. Kaguro

    An inexact differential equation

    Here, M = ##siny*cosy +xcos^{2}y ## and N = x ## M_y = (1/2)cos(2y) -xsin(2y)## and ##N_x = 1## Theorems: If R = ## \frac{1}{N} (M_y - N_x) = f(x), then I.F. = e^{ \int f(x) dx} ## If R = ## \frac{1}{M} (N_x - M_y) = g(y), then I.F. = e^{ \int g(x) dx} ## Neither is holding true. What should...
  39. PainterGuy

    I Solving a differential equation using Laplace transform

    Hi, I was trying to see if the following differential equation could be solved using Laplace transform; its solution is y=x^4/16. You can see below that I'm not able to proceed because I don't know the Laplace pair of xy^(1/2). Is it possible to solve the above equation using Laplace...
  40. karush

    MHB -m30b Convert the differential equation

    ov!347 nmh{1000} Convert the differential equation $$y''+5y'+6y=e^x$$ into a system of first order (nonhomogeneous) differential equations and solve the system. the characteristic equation is $$\lambda^2+5\lambda+6=e^x$$ factor $$(\lambda+2)(\lambda+3)=e^x$$ ok not real sure what to do with...
  41. Q

    Help to reduce solution of differential equation: dy/dx=(xy+y)/(x+xy)

    ln(y) +y = ln(x) + x +C y=?
  42. Physics345

    Differential Equation ODE Solution help.

    dM/dY = x+2y+1 dN/dx = 1 (My-Nx)/n = 1 Integrating Factor => e^∫1dx= e^x (xye^x+ye^x+ye^x)dx + (xe^x+2ye^x)dy = 0 dM/dY =xye^x+e^x+2ye^x dN/dx = xye^x+e^x+2ye^x Exact ∫dF/dy * dy = ∫ (xe^x+2ye^x)dy F = xy*e^x + y^2*e^x + c(x) dF/dx = xy*e^x + y*e^x + y^2 * e^x + c'(x)...
  43. G

    I Find the general solution for the differential equation

    So in my previous math class I spotted on my book an exercise that I couldn't solve. We had to find the general solution for the differential equation. This was the exercise: 4y'' - 4y' + y = ex/2√(1-x2) Can anyone tell me how to solve this step by step?
  44. B

    Linearizing the Lugiato-Lefever Partial Differential Equation

    I started by substituting the following anzatz: $$ \psi = \psi_e + \psi_1 $$ When ## |\psi_1| \ll 1 ##. Substituting the above into the equation yields: $$ \frac {d\psi_1} {dt} = -(1 + i\alpha)\psi_1 + \frac i 2 \frac {\partial ^ 2 \psi _1 } {\partial x ^ 2 } + i (\bar \psi_1 \psi_1 ^2 + \bar...
  45. christang_1023

    Solve the differential equation of motional emf

    . Above is the figure of the problem. I am trying to solve x(t) and differentiate it to obtain v(t); however, I have difficulty solving the differential equation shown below. $$ v(t)=\int a(t)dt=\int \frac{B(\varepsilon-Blv)d}{Rm}dt \Rightarrow \frac{dx}{dt}=\frac{B\varepsilon...
  46. F

    Differential equation modeling glucose in a patient's body

    The rate at which glucose enters the bloodstream is ##r## units per minute so: ## \frac{dI}{dt} = r ## The rate at which it leaves the body is: ##\frac {dE}{dt} = -k Q(t) ## Then the rate at which the glucose in the body changes is: A) ## Q'(t) = \frac{dI}{dt} + \frac {dE}{dt} = r - k...
  47. Celso

    Differential equation for the simple pendulum

    How do I start this? I plugged the differential equation at wolfram alpha and it semmed too complicated for such an exercise. I've also looked at a sample of an answer on cheeg where the initial approach is to rewrite the equation as ##\frac{d}{dt} (\frac{\dot\theta^2}{2}-cos(\theta)) = 0## How...
  48. Phys pilot

    I How do I classify this partial differential equation? Inhomogeneous?

    Hello, I have to solve this second order differential equation. It's like a string vibrating equation but with a constant c: $$\frac{{\partial^2 u}}{{\partial t^2}}=k\frac{{\partial^2 u}}{{\partial x^2}}+c$$ B.C $$u(0,t)=0$$ $$u(1,t)=2c_0$$ c_0 is also a constant I.C $$u(x,0)=c_0(1-\cos\pi...
  49. AbusesDimensAnalysis

    A Differential equation involving a time series

    Hey all, it's been awhile since done any calculus or DE's but was trying out some modelling (best price price per item for bulk value deals as a function of time and amount), in the last line i have f(n,t) implicitly. Any pointers or techniques for solving such things?
  50. Boltzman Oscillation

    Help derive this differential equation?

    Hello I need to derive this equation from Grittfith's quantum book $$ \frac{d^2y}{dr^2} = r^2y$$ I know I can use the characteristic equation: $$m^2 = r^2 \rightarrow y = e^{r^2}$$ but the answer should be: $$y=Ae^{\frac{-r^2}{2}} + Be^{\frac{r^2}{2}}$$ I know from Euler's formula that...
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