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

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How to solve the differential equation: $x \frac{d^2y}{dx^2}+2(3x+1)\frac{dy}{dx}+3y(3x+2)=18x$

How to solve the differential equation $$x \frac{d^2y}{dx^2}+2(3x+1)\frac{dy}{dx}+3y(3x+2)=18x$$ I think I could let $u=xy$ but I don't know how to proceed it.
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2answers
56 views

Can you find this tough Integrating Factor?

$$ u\left(du − dv\right) + v\left(du + dv\right) = 0 $$ Can you get this equation in a form by which the integrating factor can be found using the standard algorithm? Expression $\;\displaystyle \mu ...
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1answer
246 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 ...
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2answers
35 views

Could someone please verify whether or not this is a book error?

Below is a short extract for which I believe there may be an error: I think that equations $(22.3)$ and $(22.4)$ have been written out wrongly, they should be ...
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2answers
50 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$...
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0answers
23 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 ...
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0answers
43 views

$h$-principle for Legendrian immersions

It is "folklore" that continuous curves in contact 3 manifolds can be approximated by Legendrian curves and it seems that this follows from Gromov's $h$-principle for Legendrian immersions (in ...
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0answers
16 views

Bibliography for learning partial differential equations

I need to learn partial differential equations for a job. Are there any online resources (books or tutorials) good enough to learn how to solve partial differential of all kinds (homogeneous, linear &...
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4answers
71 views

Differential Equations (Coffee)

This is a long post so bear with me until I get to the part where I am stuck on! :) Question: The author of a popular detective novel drinks black coffee to help him stay awake while writing. ...
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0answers
23 views

Solution to the boundary value problem $y''+\left(\frac{(1-e^{-4x})^2}{2x^3}-\frac{2(1-e^{-4x})}{x}\right)y'-\frac{(1-e^{-4x})^4}{16x^4} y=0$

Solve the following boundary value problem $$y''+\left(\frac{(1-e^{-4x})^2}{2x^3}-\frac{2(1-e^{-4x})}{x}\right)y'-\frac{(1-e^{-4x})^4}{16x^4} y=0, \quad y(0)=y(\tfrac{1}{2})=0.$$ Note: I attempted ...
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2answers
76 views

Increasing function with $f'(x)=f(f(x))$

Is there a strictly increasing function $f: \mathbb{R}\rightarrow \mathbb{R}$ such that $f'(x)=f(f(x))$ for all $x$?
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0answers
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Reference request: how to check whether a set is invariant for a second order dynamical system?

I am looking for some examples where invariant set is proved for second order systems For a example, consider the Van Der Pol equation: $$\dot x_1 = x_2$$ $$\dot x_2 = -x_1 + 0.5(1-x_1^2)x_2$$ In ...
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1answer
18 views

First order ODE and cos(u)/sin(u) idenitity

I have a first order ODE I must put into a specific form, but this part of the given solution is a mystery to me. $\int\frac{\cos u}{\sin u}du = -\int\frac{1}{x}dx$ This can be solved so that ...
5
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1answer
30 views

Difficulty understanding step in Kac's proof of Feynman-Kac Theorem

I am trying to understand a proof of the Feynman-Kac Theorem, as set out in Mark Kac's 1949 paper 'On Distributions of Certain Wiener Functionals'. Kac defines a series of independent and ...
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2answers
33 views

Why is the integrating factor for this equation not an exponential function?

MIT Open CourseWare 18.03 Spring 10 Exercises 1B-2 c) My Question: What is the logic and method by which the correct integrating factor was found? I found an exponential function that is not the ...
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0answers
9 views

What is necessity for integral to be well-defined in defining solutions?

When trying to give a notion of solutions for differential equations with non-local terms, e.g., integral of unknown functions, to guarantee that the integral is well-defined, i.e., finitely ...
3
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1answer
69 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 ...
2
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1answer
20 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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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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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
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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 $t=r\cos{x}$...
3
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1answer
41 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....
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0answers
20 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 ...
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1answer
25 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
65 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'...
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1answer
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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? ...
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1answer
28 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: ...
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1answer
140 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
48 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 $$\frac{d^2y}{dt^2}-5\frac{dy}{dt}+6y=3+20\sin t$$
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2answers
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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
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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 ...
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1answer
79 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&...
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2answers
149 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 ...
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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 +...
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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 ...
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1answer
27 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 ...
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1answer
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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 $\...
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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/...
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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 ...
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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 ...
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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-...
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2answers
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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^{(...
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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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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 ...