Ordinary differential equation Definition and 97 Threads

TL;DR Summary: Comprehensive books on D Operator method Currently I am studying (Ordinary) Differential equation but the book I am following doesn't include much on D Operator Method of solving differential equations. Please suggest me some (Ordinary Differential Equation) books that goes...

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...

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...

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...

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...

##\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...

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##...

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...

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 =...

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...

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...

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...

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...

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...

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...

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...

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)...

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')...

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...

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...

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...

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...

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 =...

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...

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 =...

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

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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 =...

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...

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...

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...

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...

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.

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}...

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

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...

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...

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...