What is Ordinary differential equation: Definition and 96 Discussions

In mathematics, an ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and the derivatives of those functions. The term ordinary is used in contrast with the term partial differential equation which may be with respect to more than one independent variable.

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  1. P

    Solve an ODE using Fourier series

    I've assumed ##y(t)## to be the sum of a complex Fourier series, and we get $$\sum (-n^2)c_ne^{int}+\sum ac_ne^{int}=\sum c_ne^{int}e^{in\pi},$$ which we can write as $$\sum ((-n^2)+a)c_ne^{int}=\sum (-1)^n c_ne^{int}.$$ We see here that equality holds if ##a=(-1)^n+n^2##. But how do I solve...
  2. P

    Avoid unpleasant integrals in solving IVP

    The formula I'm given is that the general solution to a linear inhomogeneous system ##x'(t)=A(t)x(t)+b(t)## is ##x(t)=F(t)\int F^{-1}(t)b(t)dt##, where ##F(t)## is the fundamental matrix to the linear homogenous system (here ##A(t)## is an ##n\times n## matrix function and ##b(t)## and ##n\times...
  3. P

    Linear homogenous system with repeated eigenvalues

    I've solved this problem using a fairly involved technique, where I compute the matrix ##e^{tA}## (the fundamental matrix of the system) with a method derived from the Cayley-Hamilton's theorem. It is a cool method that I believe always works, but it can be a lot of work sometimes. It involves...
  4. P

    Solution verification of ODE using Frobenius' method

    I have no problems with solving this exercise, but my solution disagrees slightly with that given in the answers in the back of the book, and I do not know who's correct. First, we rewrite the equation as $$x''+\frac{3}{2t}x'-\frac{(1+t)}{2t^2}x=0.\tag1$$ We recognize that this is so-called...
  5. P

    On weakly singular equations and Frobenius' method

    ##\mu## is a root of the so-called indicial equation ##I(\lambda)=\lambda(\lambda-1)+p_0\lambda+q_0##. ##\nu## can also be a root of the indicial equation, or we may have ##\mu=\nu## My attempt so far is trying to characterize both equations according to Definition 2, as well as identifying...
  6. P

    Continue solutions of ODEs around the origin

    What confuses me is that my solution differs from that given in the answers at the back of the book. Solving the ODEs is fairly simple. They are both separable. After rearrangement and simplification, you arrive at ##x(z)=Cz^{1/2}## for a) and ##x(z)=De^{1/z}## for b). In both solutions, ##C##...
  7. P

    On asymptotically stable systems and bounded solutions

    We need to show ##\lVert x(t)\rVert## is bounded. It is given that ##\lVert b(t)\rVert\leq c_1## for ##t\geq t_0##. A TA has claimed that ##\lVert e^{tA}\rVert\leq ce^{-\epsilon t}## holds for some ##\epsilon>0## and a constant ##c##, when ##t\geq0##. I have a hard time confirming this claim and...
  8. G

    [Sturm-Liouville eigenvalues and eigenfunctions problem]

    I have found that w(x) should be e^-x to make L self-adjoint. and insert back get xL''+(x+1)L' +lambda L = 0 now it needs to assume a monic polynomial function, so I assume Ln = x^n+ sum from k=0 to n-1 (a_k*x^k) get the 1st and 2nd order differential and insert back I get lambda_n =...
  9. P

    Show Picard iteration diverges

    For an example of a Picard iteration, see here. In this case, we have \begin{align} &x_0(t)=x(0)=0,\nonumber\\ &x_1(t)=x_0(t)+\int_0^t \big(1+(x_0(s)-s)\big)^2ds=t+\frac{t^3}{3},\nonumber \\ &x_2(t)=x_0(t)+\int_0^t \big(1+(x_1(s)-s)\big)^2ds=t+\frac{t^7}{3^27},\nonumber\\ &\cdots \nonumber...
  10. P

    I On sub and super solutions: Teschl and others

    I'm reading Ordinary Differential Equations by Andersson and Böiers. There is a comparison theorem I have some questions about. I have also checked Teschl's Ordinary Differential Equations and Dynamical Systems, but there I have problems with his definition of a sub solution. I'll elaborate...
  11. P

    Find domain where function is Lipschitz

    The reduction is simple in all cases. For the first one, put ##x_1=x, x_2=x'## and ##x_3=x''##. Let ##\pmb{x}=(x_1,x_2,x_3)##. Then we get $$\pmb{x}'= \begin{pmatrix}x_1' \\ x_2' \\ x_3' \end{pmatrix}=\begin{pmatrix}x_2 \\ x_3 \\ 1-x_1^2 \end{pmatrix}=\pmb{f}(\pmb{x}),$$ where...
  12. P

    I On the approximate solution obtained through Euler's method

    This is a bit of a longer post. I have tried to be as brief as possible while still being self-contained. My questions probably do not have much to do with ODEs, but this is the context in which they arose. Grateful for any help. In what follows ##|\cdot|## denotes either the absolute value of...
  13. P

    I On error estimates of approximate solutions

    I'm reading Ordinary Differential Equations by Andersson and Böiers. They give an estimate for how the difference between an exact and an approximate solution propagates with time. Then they give an example where they encourage the reader to check that this estimate holds. When I do that, I get...
  14. P

    I Verifying properties of Green's function

    I'm reading about fundamental solutions to differential operators in Ordinary Differential Equations by Andersson and Böiers. There is a remark that succeeds a theorem that I struggle with verifying. First, the theorem: If the leading coefficient in ##(1)## is not ##1## but ##a_n(t)##, then...
  15. P

    Separable first order ODE involving tangent

    By inspection, we see that ##x=k\pi## is a solution for ##k\in\mathbb Z##. Moreover, the equation implicitly assumes ##x\neq n\pi/2## for odd ##n\in\mathbb Z##, since ##\tan x## isn't defined there. So suppose ##x\neq k\pi##, i.e. ##\tan x\neq 0##, then rearranging and writing ##\tan...
  16. H

    Mr. Tenenbaum's and Prof. Mattuck's advice not working (ODE)

    All right, we got $$ y'' + y = 4x \sin x $$ We are doing the Complexification $$ \tilde{y''} + \tilde{y} = 4x e^{ix} $$ Complementary function: $$ \begin{align*} \textrm{characteristic equation =}\\ m^2 + 1 = 0 \\ m = \pm i \\ \tilde{y_c} = c_1 e^{ix} + c_2 e^{-ix} \\ \end{align*} $$ Q(x)...
  17. 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')...
  18. A

    I What are the applications of function-valued matrices?

    I'm glad there's a section here dedicated to differential equations. I've seen in the fundamental theorem of linear ordinary systems, that, for a real matrix ##A##, we have ## d/dt \exp(At) = A \exp(At)##. I'm wondering if there are analogs of this, like for instance, generalizing a system of...
  19. H

    I Analysis of converting a DE into complex DE

    In Lecture 7, Prof. Arthur Mattuck (MIT OCW 18.03) taught that the following equation $$ y’ +ky = k \cos(\omega t)$$ can be solved by replacing cos⁡(ωt) by ##e^{\omega t}## and, rewriting thus, $$ \tilde{y’} + k\tilde{y}= ke^{i \omega t} $$ Where ##\tilde{y} = y_1 + i y_2##. And the solution of...
  20. A

    I Non-linear ODE: initial conditions

    Say you have the set of coupled, non-linear ODEs as derived in this thread, it has two unknowns ##N(t)## and ##\theta(t)##: $$ N - mg = - m\frac{L}{2}\left(\dot{\theta}^2\cos(\theta) + \ddot{\theta}\sin(\theta)\right)$$ $$ \frac{L}{2}N\sin(\theta) = \frac{1}{12}ml^2\ddot{\theta}$$ What freedom...
  21. A

    Vertical beam on a frictionless surface

    This could also be posted in the Math / differential equations sub, but it also involves the derivation which is classical physics. So I was doubting :smile:. So, I'm dusting off my dynamics a bit and found this problem of a thin beam on a frictionless surface in a different forum and decided...
  22. Athenian

    Rowing a Boat Across a River - FODE Physics Problem

    First off, to solve for this problem, I relied largely on my below drawn diagram. Forgive the poor work as this was done on a laptop. Using the above image as reference, I came up with the below equations. $$\frac{dy}{dx} = \frac{sin \, \theta}{1- cos \, \theta}$$ where ##cos \, \theta =...
  23. J

    Why does the Euler approximation fail for the Airy or Stokes equation?

    I had thought it would be failure of structural stability since in structural stability qualitative behavior of the trajectories is unaffected by small perturbations, and here, even tiny deviations using ##h## values resulted in huge effects. However, apparently that's not the case, and I'm not...
  24. 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 =...
  25. T

    A Determine PDE Boundary Condition via Analytical solution

    I am trying to determine an outer boundary condition for the following PDE at ##r=r_m##: $$ \frac{\sigma_I}{r} \frac{\partial}{\partial r} \left(r \frac{\partial z(r,t)}{\partial r} \right)=\rho_D gz(r,t)-p(r,t)-4 \mu_T \frac{\partial^2z(r,t)}{\partial r^2} \frac{\partial z(r,t)}{\partial t} $$...
  26. S

    I Y'' + y = 0 solution and recursion relation

    I've found the general solution to be y(x) = C1cos(x) + C2sin(x). I've also found a recursion relation for the equation to be: An+2 = -An / (n+2)(n+1) I now need to show that this recursion relation is equivalent to the general solution. How do I go about doing this? Any help would be...
  27. Martin T

    I About Arnold's ODE Book Notation

    In Arnold's book, ordinary differential equations 3rd. WHY Arnold say Tg:M→M instead of Tg:G→S(M) for transformations Tfg=Tf Tg, Tg^-1=(Tg)^-1. Let M be a group and M a set. We say that an action of the group G on the set M is defined if to each element g of G there corresponds a...
  28. Martin T

    Vladimir I. Arnold ODE'S book, about action group

    hi everyone, I'm electrical engineer student and i like a lot arnold's book of ordinary differential equations (3rd), but i have a gap about how defines action group for a group and from an element of the group.For example Artin's algebra book get another definition also Vinberg's algebra book...
  29. Peter Alexander

    Solving Partial Differential Equation

    1. The problem statement, all variables, and given/known data Task requires you to solve a partial differential equation $$u_{xy}=2yu_x$$ for ##u(x,y)##. A hint is given that a partial differential equation can be solved in terms of ordinary differential equations. According to the solution...
  30. L

    I Second order ordinary differential equation to a system of first order

    I tried to convert the second order ordinary differential equation to a system of first order differential equations and to write it in a matrix form. I took it from the book by LM Hocking on (Optimal control). What did I do wrong in this attachment because mine differs from the book?. I've...
  31. A

    I How to find a solution to this linear ODE?

    I want to find solution to following ODE $$ \frac{d \bar h}{dt} + \frac{K}{S_s} \alpha^2 \bar h = -\frac{K}{S_s} \alpha H h_b(t) $$ I have solved it with integrating factor method with ## I=\exp^{\int \frac{1}{D} \alpha^2 dt} ## as integrating factor and ##\frac{K}{S_s} = \frac{1}{D} ## I have...
  32. A

    A How to simplify the solution of the following linear homogeneous ODE?

    During solution of a PDE I came across following ODE ## \frac{d \bar h}{dt} + \frac{K}{S_s} \alpha^2 \bar h = -\frac{K}{S_s} \alpha H h_b(t) ## I have to solve this ODE which I have done using integrating factor using following steps taking integrating factor I=\exp^{\int \frac{1}{D} \alpha^2...
  33. G

    Wondering if these two First Linear Order IVPs are correct

    Homework Statement I am having trouble proving if the equation i have found for number 1 is correct. I have posted my solution to get back to the main problem in the first photo below. For number 2 I am having trouble isolating for 1 y(x). Did i do the integration and setup properly?Homework...
  34. B

    Linear ordinary differential equation.

    Homework Statement ##\dfrac{dy}{dx} + y = f(x)## ##f(x) = \begin{cases} 2 \qquad x \in [0, 1) \\ 0 \qquad x \ge 1 \end{cases}## ##y(0) = 0## Homework EquationsThe Attempt at a Solution Integrating factor is ##e^x## ##e^x\dfrac{dy}{dx} + e^x y = e^x f(x)## ##\displaystyle ye^x = \int e^x...
  35. Poetria

    Basis for the space of solutions (ODE)

    Homework Statement The equation given: dy/dt = 3*y A basis for the space of solutions is required.The Attempt at a Solution According to me it is e^(3*t) but it has turned out false. Why? I am considering the answer "The basis is the set of all functions of the form c*e^(3*t) but a...
  36. Mayan Fung

    Introducing SHM to high school students

    I need to introduce Simple Harmonic Motion to a group of high school students studying physics. They don't know anything about differential equation except the method of separation of variables. Also, they have limited knowledge on complex numbers like eiωt. However, I don't want to just give...
  37. kostoglotov

    Is there a mistake in the assignment?

    Homework Statement I'm actually a tutor, and a student of mine at uni has the following differential equation with initial conditions to solve imgur link: http://i.imgur.com/ptuymQv.gif From y(t) = c_1sin(3t) + c_2cos(3t), it is not possible to solve for the constants using the given...
  38. samg1

    Ordinary Differential Equation - tank with inflow and outflo

    Homework Statement A tank contains 60 kg of salt and 2000 L of water. A solution of a concentration 0.015 kg of salt per liter enters a tank at the rate 6 L/min. The solution is mixed and drains from the tank at the same rate. Find the amount of salt in kg at t = 3 hours Find the...
  39. Nipuna Weerasekara

    A non-exact nonlinear first ODE to solve

    Homework Statement Solve the following equation. Homework Equations ( 3x2y4 + 2xy ) dx + ( 2x3y3 - x2 ) dy = 0 The Attempt at a Solution M = ( 3xy4 + 2xy ) N = ( 2x3y3 - x2 ) ∂M/∂y = 12x2y3 + 2x ∂N/∂x = 6x2y3 - 2x Then this equation looks like that the integrating factor is (xM-yN). IF =...
  40. Summer95

    The Dirac Delta Function

    Homework Statement Differential equation: ##Ay''+By'+Cy=f(t)## with ##y_{0}=y'_{0}=0## Write the solution as a convolution (##a \neq b##). Let ##f(t)= n## for ##t_{0} < t < t_{0}+\frac{1}{n}##. Find y and then let ##n \rightarrow \infty##. Then solve the differential equation with...
  41. F

    I General solution to linear homogeneous 2nd order ODEs

    Given a linear homogeneous 2nd order ODE of the form $$y''(x)+p(x)y'(x)+q(x)=0$$ the general solution is of the form $$y(x)=c_{1}y_{1}(x)+c_{2}y_{1}(x)$$ where ##c_{1},c_{2}## are arbitrary constants and ##y_{1}(x), y_{2}(x)## are linearly independent basis solutions. How does one prove that...
  42. G

    Find a function with given condition

    Homework Statement Find a curve that passes through point A(2,0) such that the triangle which is defined with a tangent at arbitrary point M, axis Oy and secant \overline{OM} is isosceles. \overline{OM} is the base side of a triangle. 2. The attempt at a solution Function passes through point...
  43. P

    Application of boundary conditions in determining the Green's function

    Homework Statement Find the Green's function $G(t,\tau)$ that satisfies $$\frac{\text{d}^2G(t,\tau)}{\text{d}t^2}+\alpha\frac{\text{d}G(t,\tau)}{\text{d}t}=\delta(t-\tau)$$ under the boundary conditions $$G(0,\tau)=0~~~\text{ and }~~~\frac{\text{d}G(t,\tau)}{\text{d}t}=0\big|_{t=0}$$ Then...
  44. J

    Applied Any books on Ordinary Differential Equations w/ applications

    Hello, I really need a good book on ordinary differential equations with applications on Quantum Mechanics, as I will be attending a course on QM but I do not have the proper mathematical background that is needed.
  45. Linder88

    Ordinary differential equation with boundary value condition

    Homework Statement Consider the boundary value problem \begin{equation} u''(t)=-4u+3sin(t),u(0)=1,u(2)=2sin(4)+sin(2)+cos(4) \end{equation} Homework Equations Derive the linear system that arise when discretizating this problem using \begin{equation} u''(t)=\frac{u(t-h)-2u(t)+u(t+h)}{h^2}...
  46. 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
  47. Linder88

    Laplace transform of y''(t')

    The ordinary differential equation, with initial values,shall be solved using Laplace transform. The ODE looks like this \begin{equation} y''(t')+2y''(t)-2y(t)=0 \end{equation} And the initial conditions are \begin{equation} y(0)=y'(0)=0, y''(0)=0 \end{equation} The problem is with the first...
  48. P

    Matrix-free iteration methods and implicit ODE solvers

    Im trying to implement the implicit Euler method in high-performance software for micromagnetic simulations, where I'm restricted to using the driving function of the ODE (Landau-Lifshitz equation) and the previous solution points. This obviously not a problem for an explicit method, since we...
  49. VoteSaxon

    Finding the Particular Integral for d2y/dt2 + 4y = 5sin2t

    Homework Statement Well I am looking for the particular integral of: d2y/dt2 + 4y = 5sin2t The attempt at a solution As f(t) = 5sin2t, the particular integral yPI should look like: yPI = Acos2t + Bsin2t dyPI/dt = -2Asin2t + 2Bcos2t d2yPI/dt2 = -4Acos2t - 4Bsin2t Subbing into the differential...
  50. Mark Brewer

    Ordinary Differential Equation Problem

    Homework Statement dy/dx = 4e-xcosxThe Attempt at a Solution [/B] I've divided dx to both sides, and now have dy = 4e-xcosx dx I've then started to use intergration by parts to the right side with u = 4e-x and dv = cosx dx Leaving y = 4e-xsinx - ∫ -4e-xsinx dx Once again I used intergration...
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