Questions on (ordinary) differential equations. For questions specifically concerning partial differential equations, use the (pde) tag.

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1answer
24 views

How to get the correct angle of the ellipse after approximation

I need to get the correct angle of rotation of the ellipses. These ellipses are examples. I have a canonical coefficients of the equation of the five points. $$Ax ^ 2 + Bxy + Cy ^ 2 + Dx + Ey + F = 0$...
2
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1answer
17 views

What is the necessary condition for ODE to have unique solution?

For the ODE: \begin{align} \dot{x}(t)&=f(x,t) \\ x(t_{0})&=x_{0} \end{align} If $\;\;f:\mathbb{R}^{n}\rightarrow{}\mathbb{R}^{n}$ is Lipschitz continuous on $\mathbb{R}^{n}$, then there exists ...
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0answers
10 views

Showing that ODE is not of Sturm-Liouville form

The PDE $$\frac{\partial u}{\partial t} = D\frac{\partial^2 u}{\partial x^2}-V_0\frac{\partial u}{\partial x}$$ can be separated into two ODEs by the method of separation of variables, and the ODE ...
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0answers
16 views

Find solution of Laplace equation

Hey I need help with these example: Solve boundary problem on $\mathbb{R}^{+} \times \mathbb{R}^{+}$ \begin{equation*} \left\{ \begin{array}{l} \Delta u = 0 \\ u(0,.)=0 \\ u(.,0)= f\end{array}\right....
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0answers
16 views

Eigenfunctions of $x^2M''+xM'+\lambda M=0$ with $M'(1)=0$ and $M'(L)=0$

If we make the substitution of variables by $z=\ln(x)$ in $$x^2M''+xM'+\lambda M=0$$ then we will get $$M''(z)=-\lambda M(z)$$ We can consider different cases for $\lambda$: Case 1: $\lambda>0$ ...
3
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2answers
76 views

Solve $\frac{dx}{dt}=\frac{at-\cos{x}}{at^2\tan{x}+t}$

Solve $\begin{align*}\frac{dx}{dt}=\frac{at-\cos{x}}{at^2\tan{x}+t}\end{align*}\\\\ $ Am I justified in doing the following substitution? If not, can a closed-form solution be found? Let $t=r\cos{x}$...
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1answer
8 views

Solving a system of N-1 Ist order ODEs by Euler's Method

In order to solve a system of N-1 first order ODEs by Euler's Method For N = 4; t=0, h= 0.1, x= 0.1 should the Euler formula be? $U_n(t+h) = U_n(t) + h F_n(x_n, t_n)$ for n = 1, 2,..,N-1 but we ...
4
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1answer
37 views

The differential equation $y'(x)=\lambda \sin(x+y(x)),y(0)=1.$

For $\lambda\in\mathbb{R},$ consider the differential equation $$y'(x)=\lambda \sin\left(x+y(x)\right),y(0)=1.$$ Then the initial value problem has: $1.$ no solution in any neighbourhood of $0.$ $2....
2
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0answers
19 views

How to reflect this phase plane?

This is the phase plane of $X'(t)=\bigl(\begin{smallmatrix} 3 &1 \\ -4 &-1 \end{smallmatrix}\bigr)X(t)$. Which ODE system correspond to a phase plane that is a reflected image along the ...
1
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1answer
24 views

The differential equation $-y''+(1+x)y=\lambda y,x\in (0,1).$

The problem $$-y''+(1+x)y=\lambda y,x\in (0,1), y(0)=y(1)=0$$ has a non zero solution $1.$ for all $\lambda <0.$ $2.$ for all $\lambda\in[0,1].$ $3.$ for some $\lambda\in (2,\infty).$ $4.$ for ...
3
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1answer
61 views

Solve the non-linear differential equation

I have been trying to solve the following differential equation: $$ \dot{y} = \frac{3x^2}{y-x^2+1}$$ Substituting $u=y-x^2+1$ we get $\dot{u}=\dot{y}-2x$ we get $\dot{u}=\frac{3x^2}{y}-2x$. But I can'...
3
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1answer
52 views

Can I find a function of $y$ that satisfies the relation $\dfrac{df(y)}{dx} = y^2(3y'+1)$

Suppose we have an unknown function $y=y(x)$ , is it possible to find a function $f(y)$ such that: $$\dfrac{df}{dx}= y^2\left(3\dfrac{dy}{dx}+1 \right)$$? EDIT: of course if there is no $1$ in the ...
0
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1answer
18 views

Advice about formula for exact differential equation

When I realize that I have exact differential equation I know that is good to use specific formula. But this formula has two forms. Can you tell me when I must use first and when second form? ...
1
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1answer
27 views

Solving Ordinary Differential Equations Using Derivative with Respect to Time

I am trying to solve this ODE. But I only have the following parameter values: ...
0
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1answer
138 views

Is there an english edition of Jorge Sotomayor's book on differential equations?

I am currently using "Lições de equações diferenciais ordinárias", in portuguese, by Jorge Sotomayor. However portuguese is not my best language by a long shot, and I struggle a little. Does anyone ...
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4answers
44 views

$x^2\frac{d^2y}{dx^2}-4x\frac{dy}{dx}+6y=3+20\sin ({\ln x})$

How to show that the substitution $x=e^t$ transforms the differential equation $$x^2\frac{d^2y}{dx^2}-4x\frac{dy}{dx}+6y=3+20\sin ({\ln x})$$ into $$\frac{d^2y}{dt^2}-5\frac{dy}{dt}+6y=3+20\sin t$$
1
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2answers
75 views

Integral solution of separable differential equation

On page 524 of Tenenbaum's Introduction to Analytic and Probabilistic Number Theory (3rd edition) it is essentially stated that the solution to the first-order differential equation $$y' = e^{-x}y/x \...
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1answer
28 views

2nd order differential equation with limits

Solve the differential equation $\frac{d^2y}{dx^2}-2\frac{dy}{dx}-3y=2e^{-x}$ given that $y\rightarrow0$ as $x\rightarrow \infty$ and that $\frac{dy}{dx}=-3$ when $x=0$ My attempt, I've already ...
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0answers
12 views

dominated theorem

If $\phi(t)$ and $\psi(t)$ are fundamental matrices of differential equations $ dX(t)=A(t)X(t)dt$ and $ dX(t)=B(t)X(t)dt $ If $ g(t)$ is a bounded and measurable function then is it correct to ...
0
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1answer
72 views

Why can this differential equation be written in $3$ different ways?

Suppose we have the following differential equation using operator notation: $$(D-x)(D+x)y=0\tag{1}$$ where $$D=\frac{d}{dx}$$ Now I could rewrite $(1)$ as $$\begin{align}\require{enclose}(D-x)(D+x)y&...
3
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2answers
148 views

Application of Bessel Function

I have read number of books and online literature on Bessel function. Theoretically, I have known about Bessel function. What is practical significance of Bessel function? How can Bessel function ...
1
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1answer
26 views

Dampened mathematical pendulum

I have the system of ODEs $$\begin{align*}\dot y &= v \\ \dot v &= -\lambda v - q(y) \end{align*}$$ for an increasing function $q$ such that $q(0)=0$ and the energy function $E= \frac 1 2 v^2 +...
1
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1answer
394 views

Showing Schlaefli integral satisfies Legendre equation

The integral representation of Legendre functions is $P_ν(z)=\frac{2^{-\nu}}{2\pi i} \oint_Γ\frac{(w^2−1)^\nu}{(w−z)^{\nu+1}} dw$. I'm trying to show that this satisfies Legendre's equation. When I ...
3
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1answer
225 views
+50

Deriving the Airy functions from first principles

I have just started reading about the Airy functions and am stuck on a particular step of their derivation. But first here is some background information to give this question some meaning, more ...
0
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1answer
26 views

solving a differential equation using substitution

ok so I have to use the substitution x=e^t to change the following DE $$x^2y''-3xy'+y = 1+ x^8 \ln^3 x +xe^{5x}$$ into a linear DE, I also have prove the needed chain rule, any help?? all I know in ...
0
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1answer
17 views

Lyapunov function guarantees local exponential stability

can someone give me a proof of http://nptel.ac.in/courses/101108047/module13/Lecture%2031.pdf page 15? Suppose all conditions for asymptotic stability are satisfied. In addition to it, suppose $\...
0
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1answer
194 views

Reducible to Separable First Order Differential Equation Word Problem in Analytic Geometry 1.4-29

I completed near all problems om a differential equations text chapter on reducing non-separable first order differential equations to separable by using an appropriate substitution for example $u = y/...
1
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3answers
368 views

Tricky Separable Differential Equation

Please guide me: $y' + ay +b = 0$ ($a$ not zero) is supposed to be separable and has solution $y = ce^{-ax} - \frac ba$ Here is my start to this problem: $\frac{dy}{dx} + ay = -b$ is as far as I ...
2
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1answer
40 views

Stability of a line of equilibria

I'm working with a nonlinear autonomous system $x'=f(x)$. This system stays in $\mathbb{R}^n_+$ whenever it begins there, and it has a ray of equilibria, i.e. there is a positive vector $x_0$ so that ...
2
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2answers
49 views

Method of Annihilators Tedium…

One of the exam preparation questions for MIT's online Honors Differential Equations course asks for a general solution of \begin{align} (D^2 - 1)^4(D^3 + 1)^5y = 3e^t \end{align} The fact that the ...
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1answer
394 views

Runge Kutta Method Matlab code

So I have a programming assignment with the following instructions: Consider the nth-order differential equation $$Ax^n (t) = x ^{(n-1)}(t) + x^{(n-2)}(t) + ... + x(t)$$ where $A$ is a real-...
1
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2answers
52 views

Rewriting ODE in terms of a different variable ($z=e^x$)

Given the ODE $$x^2M''+xM'+\lambda M = 0$$ where $1<x<L$, with boundary conditions $M(1) = 0$, $M(L)=0$, we can rewrite it in the Sturm-Liouville form and get $$\left[M'\exp\left(\int\limits_0^L{...
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1answer
39 views

Exponential matrix definition

The exponential matrix is $e^{tA} : = X(t)$ where $X$ is the unique global solution of the differential equation $x'=Ax$ which satisfies $X(0)=I$. I want to prove, using this definition, that $$ e^{(...
0
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1answer
48 views

Analytical solution for a Non-linear differential equation $\frac{d^2y}{dt^2} = A\left(\frac{dy}{dt}\right)+B \sin(2Cy)$

Analytical solution for a non-linear differential equation: $\frac{d^2y}{dt^2} = A \left(\frac{dy}{dt}\right)+ B \sin(2Cy)$ A,B are non-zero constants and y (position) is a scalar-value parameter ...
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0answers
57 views

Analytical solution for a Non-linear differential equation $\frac{d^2y}{dt^2} = A\left(\frac{dy}{dt}\right)+[B \sin(Cy)\times\cos(Dt)]-E \sin(2Cy)$

Is there any analytical solution for the following differential equation? $\frac{d^2y}{dt^2} = A\left(\frac{dy}{dt}\right)+[B \sin(Cy)\times\cos(Dt)]-E \sin(2Cy)$ A,B,C,D are non-zero constants and ...
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0answers
25 views

Solving n-dimensional first order linear pde

While working on a problem in game theory, I'm stuck at a problem which requires me to solve the following linear first order PDE on $K$ independent variable: $\sum_{k=1}^K(\frac{\partial u}{\partial ...
0
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1answer
45 views

General solution of linear ODE

What is the most general solution of $\frac{\mathrm{d}y}{\mathrm{d}x}+Py=Q$ where $P,Q$ are constants and $u,v$ are two particular solutions? How do I proceed from here? $ye^{Px}=Q\int (e^{Px})\,dx$...
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1answer
73 views

Prove that $\exp(A+B)=\exp(A)\exp(B)$ iff $[A,B] = 0$

I have searched throughout the forum and online as well, and I got that with condition of $[A,B]=0$, $e^{(A+B)t}=e^{At}e^{Bt}$. Now the question is, to show for any matrices $A$ and $B$, it is true ...
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0answers
14 views

Show that $f$ doesn't has peryodic orbits.

Let $f:\mathbb{R}^{n}\to\mathbb{R}^{n}$ a vector field for which exist a Liapunov function $V:\mathbb{R}^{n}\to\mathbb{R}$ define over all phase space. Show that $f$ doesn't has periodic orbits. I ...
1
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2answers
33 views

Inital value problem, differential equations

I'm asked to solve the problem: $7\frac{dy}{dt}+y= 28t$ with $y(0)=2$ When I integrated I ended up getting $y= (14t^2+C)/(7+t)$, then I plugged in 0 for $t$ and 2 for $y$ to find that $C$ is 14. ...
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0answers
31 views

A vector field in a star shaped set

I'm having problems trying to proof Poincaré's lemma for Star-Shaped sets Let $F:U\to \mathbb R^{2}$ be $C^{1}$ functions,where $U\subset \mathbb R^{2}$ is a Star-shaped set. If $F_x:U\to \mathbb ...
1
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2answers
49 views

A simple second order ODE

This might be a very naive question, but is there a solution for a second order ODE of the form $$y''(x) = f(x)y(x)$$ where $f(x)$ is a general function? Any information is appreciated. Thanks.
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0answers
32 views

Does any Riccati equation admit local solutions?

Is it true that any differential equation of the form: $$w'(t)=a(t)w^2(t)+b(t)w(t)+c(t),$$ where $a,b,c:[\alpha,\beta]\to\mathbb{R}$ are continuous functions, admit a solution in the neighbourhood ...
0
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3answers
55 views

economics using differential equations

I need help with this calculus problem The producer of a certain commodity determines that to protect profits, the price p should decrease at a rate equal to half the inventory surplus $S−D$, where $...
0
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0answers
22 views

Functional differential equation separatrix

I've been spinning my wheels with the following differential equation, and would greatly appreciate any guidance on ways to attack it. I have $u(x) \geq 0$ for all $x$. Further, $x \geq 0$. The ...
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0answers
43 views

Calculus: Derivative of a summation and dot product

I'm trying to implement a speed boost to an eye-tracking algorithm (found here: http://www.inb.uni-luebeck.de/publikationen/pdfs/TiBa11b.pdf). I need to take the derivative of the eye-tracking ...
0
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1answer
43 views

Differential equation without analytic solution - comparative statics

I am facing a differential equation - with boundary condition $v(T)$ given - without an analytic solution but still need to understand how the solution is affected by a change of the function's value. ...
0
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0answers
22 views

Obtaining error between exact and finite element solution of a PDE when exact solution is not available

How does one obtain the error between the finite-element (FE) solution and the exact/analytical solution when the latter in not available? After all, isn't the purpose of a numerical method to find ...
0
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1answer
49 views

Hamiltonian differential equation involving complex logarithm

Consider the differential equation $$\begin{pmatrix}\dot p \\ \dot q \end{pmatrix} = \frac{1}{p^2+q^2}\begin{pmatrix} p \\ q \end{pmatrix}$$ where $(p,q)^T\in \mathbb R ^2 - \{0\}$. I want to show ...