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

learn more… | top users | synonyms (1)

4
votes
0answers
39 views

when does a partial differential equation have unique solution?

The differential equation $ xu_x + yu_y = 2u$ satisfying the initial conditions $y = xg(x), u=f(x)$ with $f(x) = 2x, g(x) = 1$, has no solution $f(x) = 2x^2, g(x) =1$, has infinite number of ...
4
votes
0answers
73 views

Question about solutions of $x''+(1+r(t))x=0$ when $\int_1^\infty |r(t)| dx <\infty$ .

Let $x''+(1+r(t))x=0$ where $r(t)$ is continous and $\int_1^\infty |r(t)| dx <\infty$ show that the equation has solutions $\phi_1$ and $\phi_2$ such that $$\lim_{t\to\infty} ...
4
votes
0answers
51 views

Confusion about superposition principle of the PDE and Boundary Condition of an ODE.

I want to solve a PDE like this: $\frac{\partial y}{\partial t}=a\frac{\partial ^2y}{\partial x^2}-b\frac{\partial y}{\partial x}-c y,(a,b,c\in \mathbb{R})\tag{1}$ with the boundary conditions: $ ...
4
votes
0answers
38 views

Help with an ODE

I need some help, I have this ODE but can't solve it for $y(x)$, I try every method I know, but with no succes,please, somebody can help me? $(\varepsilon-x)y=y'(-x+y^2-2x^2)$ Thanks.
4
votes
0answers
58 views

How to solve this complicated ordinary differential equation?

Consider the following non-linear ODE: $$x^2 \frac{dy}{dx} + \exp{\left(x \, \frac{d^2 y}{dx^2} \right)} = \sin \left(\frac{d^3y}{dx^3} \, \cos \left( \frac{d^4y}{dx^4} \right) \right) $$ I have no ...
4
votes
0answers
59 views

A second order differential equation

How does one solve the following differential equation $y^{"}+xy^{'}+(1-x^2)y=y\sin x$? I don't know how to proceed?
4
votes
0answers
90 views

Rolling parabola & catenary

By rolling a rigid catenary on a straight line one obtains the locus of its center of curvature as a parabola. This is well known as the natural equation connecting arc length and radius of curvature ...
4
votes
0answers
58 views

Solution for $\frac{a}{x} = \int_0^1 \frac{f(z)}{\left(f(x)+f(z)\right)^2} dz$

I am looking for the function $f(x)$ that solves $\frac{a}{x} = \int_0^1 \frac{f(z)}{\left(f(x)+f(z)\right)^2} dz$ such that $f(0)=0$. Even hints how to approach to this question would be very ...
4
votes
0answers
76 views

How to solve $\frac{dy}{dx} = \frac{x^2-y^2}{x^2(y^2+1)}$

I tried to solve this using the solution of a first order differential equation but I don't think this can be reduced to that form. How to approach this problem and find $y$? Please help.
4
votes
0answers
111 views

Operators such that $\langle Ax,x \rangle=-\langle x,Ax \rangle$

Let $X$ be a Banach space. We consider the differential equation: $$x'(t)=Ax(t), \ \ \ t\in\mathbb{R}$$ where $A$ is a bounded operator on $X$. If $X$ is a Hilbert space, and $x(t)$ is a solution of ...
4
votes
0answers
88 views

Solving second order nonlinear ODE

Having the following second order ordinary differential equation: $$ \ddot{x} = a \cos(x) $$ where, $a$ is a constant. What's an approach to solve this kind of equation?
4
votes
0answers
38 views

Wronskian different from zero and solutions of ODE.

Let $a_0, \ldots , a_{n-1}$ continuous functions in an interval $I$.Consider the equation $$x^{(n)} = a_{n-1}(t)x^{(n-1)}+\cdots+a_0(t)x. \tag 1$$ Let $\phi_1, \phi_2, \ldots,\phi_n$ $n$ are ...
4
votes
0answers
80 views

Scalar Autonomous Differential Equation?

What precisely is a scalar autonomous differential equation? I'm confused about what this precisely means, more so because we did not discuss this in any lectures nor is it, as far as I can tell, ...
4
votes
0answers
77 views

Determining whether the origin is an attracting fixed point for a scalar system

I have been asked to determine and prove the attraction properties of a continuous-time dynamical system, generated by the ODE \begin{equation} \frac{dx}{dt} =-x \end{equation} which gives the system ...
4
votes
0answers
73 views

Quadratic system of ODEs

I have a quadratic ODE system that looks like this: $\dot{x}=Ax+diag(x)Nx$ where $x \in R^n$ and $A,N \in R^{n \times n}$ and $diag(x) \in R^{n \times n}$ is a diagonal matrix in which $x$ is its ...
4
votes
0answers
145 views

Prove that all the solutions of (2): $\frac{dy}{dt}=A(t)y+f(t)$ are bounded in $ \left[t_0,+\infty \right )$

I have a problem: Assume that system (1): $$\dfrac{dx}{dt}=A(t)x$$ is stable, where $A(t) \in C\left [t_0,+\infty \right )$, when $t \to \infty$ and $$\begin{cases} & \mathrm{ } ...
4
votes
0answers
59 views

Can Fredholm integral equation of the first type be represented as a differential equation?

Can Fredholm integral equation of the first type be represented as a differential equation? In other words, given a Fredholm integral equation of the second type does there exist a differential ...
4
votes
0answers
120 views

Help me understand this differential equation solution

I found a differential equation in an old paper, where the solution is a bit hard to understand. Given this equation: $$\frac{1}{2} r^2 \left(\frac{d \phi}{dr}\right)^2 + c^2 \left(r ...
4
votes
0answers
45 views

Analog of Picard's theorem for Fractional Differential equations.

I need an analog of Picard's theorem of existence and uniqueness of solutions. The theorem is to be applied to linear fractional order differential equations with constants coefficients. I don't want ...
4
votes
0answers
126 views

Bifurcation in 3 dimensions (simple)

I am Doing a project i have a toy system that describes a bifurcation in 3 dimensions i am posting this in part because i can no longer understand what i have written down ( its been awhile) i have ...
4
votes
0answers
200 views

Heat Kernel Asymptotics on Manifold with Boundary

On a closed Riemannian manifold $M$, the heat kernel $k_t(x, y)$ of the Laplace-Beltrami operator (or more general of any generalized symmetric Laplace-type operator acting on sections of a vector ...
4
votes
0answers
166 views

The simplest delay differential equation

I am trying to understand a bit about solutions of delay differential equations, so I tried analyzing one of the most simple ones: $$u'(t)=-\beta u(t-1), \text{and for } t\in [-1,0), u(t)=\phi(t), ...
4
votes
0answers
322 views

Solving inhomogenous bessel equation

I have the following differential equation to be solved $\dfrac{d^2\psi}{dr^2}+\dfrac{d\psi}{rdr}+4\left(\omega^2-k_0^2-\dfrac{n^2}{r^2}\right)\psi=AJ_n^2(kr)+\dfrac{k}{r}J_n(kr)J_{n+1}(kr)-\omega ...
4
votes
0answers
428 views

Confused by a proof in Rudin *Functional Analysis*

I am reading Rudin's Functional Analysis and got quite confused by his proof of Thm 8.5, that is, the existence of fundamental solutions for differential operator $P(D)$, where $P$ is a polynomial. ...
4
votes
0answers
95 views

$u''+\frac{4}{x+1}u'+\frac{2}{\left(x+1\right)^{2}}u=0$ variational solution

This is a concept solution scheme derived from a particular example that I have not been able to generalise sufficiently. The objective is to find a particular solution to a certain second-order ...
4
votes
0answers
2k views

Is it possible to have Wronskian=0 with independent solutions to a linear differential equation?

In Wikipedia it says that if the Wronskian of two function is 0 everywhere it does not imply they are linearly dependent. However, in books treating differential equations it seems that, if the two ...
4
votes
0answers
234 views

A solution of $-y'' + q(x)y= \lambda y$

Could you help me with the following problem (from Poschel and Trubowitz)? I am looking for a solution of the differential equation $-y'' + q(x)y= \lambda y$, for $0 \leq x \leq 1$ with ...
4
votes
0answers
104 views

Steady-state of `degenerate' delayed differential equation

Consider the simple delayed differential equation: $X'(t) = -a X(t) + a X(t - d)$ where $d$ and $a$ are positive constants. I'm interested in the possible steady-state (stationary) solutions of ...
4
votes
0answers
173 views

How to analysis the stability of these ODE?

Study whether the null solution of the system: $$\begin{cases} \frac{dx_1}{dt}=x_2(t)\\ \frac{dx_2}{dt}=-w(t)^2 x_1(t)\\ \end{cases} $$ is Lyapunov stable, where $$ w(t)= \begin{cases} 0.4 ...
4
votes
0answers
93 views

Efficiently solving a large, sparse linear system $M(s)ab(s)=c(s)$ (determined by smooth functions) over some range of $s$

I'm looking at a differential equation on the edges of a graph (the application is neuroscience), and the Laplace transform of the solution on most of the edges has a general solution more-or-less of ...
4
votes
0answers
591 views

Finding the modified Green function for the Helmholtz equation

I've been wrestling with this question for quite some time now, and the result was like 20 leaves of paper packed with scribbling...anyway, here's the question: I need to find the solution to the ...
3
votes
0answers
23 views

Uniqueness of Rectifying Coordinates: Question for Arnold's ODE Book

In section 7 of his book Ordinary Differential Equations, VI Arnold explains the `rectification theorem', that, given an ordinary differential equation $$\dot{\mathbb{x}} = \mathbb{v(x)}$$ where ...
3
votes
0answers
22 views

Solve $x(x-1)y''+6x^2y'+3y=0$ using Frobenius's Method

Solve $x(x-1)y''+6x^2y'+3y=0$ using Frobenius's Method I can't solve this ODE. How can I get first two term? and indicial equation is also very confusing. I can solve two term recurrence ...
3
votes
0answers
68 views

What type of equation is this?

Is this equation an ODE or PDE $$ \frac{d^3u}{dx^3}−αxu=0, x∈R $$ The only thing given is $\int_R u(x) =\pi $ and $α>0$ is some constant. I have to find the solution using fourier ...
3
votes
0answers
80 views

solution uniqueness of an algebraic system

$A(v),B(v),C(v)$ are positive, convexly decreasing functions on $\mathbb{R_+}$; $x$ is a random variable that obeys distribution F; Function $v(a,b,x)$ is implicitly defined as ...
3
votes
0answers
16 views

Is there an analytic solution for this Fokker-Planck equation?

The Fokker-Planck equation for a probability distribution $P(\theta,t)$: \begin{align} \frac{\partial P(\theta,t)}{\partial ...
3
votes
0answers
63 views

If $y'=\dfrac{1}{x+1}$ and $y(0)=0$, find the value of $y(-2) $

If $y'=\dfrac{1}{x+1}$ and $y(0)=0$, find the value of $y(-2) = ?$ By integrating I am getting $$y = \ln (x+1)+C$$ I am stuck somewhat as it looks tricky from here. Any help ? Thanks!
3
votes
0answers
58 views

Solving a differential equation with a square root

I am trying to solve the differential equation $ A(x)\frac{d^{2}f(x)}{dx^{2}}+B(x)\frac{df(x)}{dx}=\frac{1}{3}\frac{1}{\sqrt{f(x)}}, $ where $ A(x)=\frac{x}{x+1} $ and $ B(x)=\frac{2x+1}{(x+1)^{2}} ...
3
votes
0answers
64 views

Examples on conceptual problems for eigenvalues in differential equations

I am currently holding a discussion class on diff eqs for engineers and I am looking for an interesting conceptual problem on eigenvalues in diff eqs. Most of the problems in 5 different books that I ...
3
votes
0answers
75 views

Wave equation for a string nonuniform (PDE)

I have tried to solve this exercise from Applied Partial Differential Equations-Richard Haberman , but I have been impossible these paragraphs. The displacement $u$ of a nonuniform string ...
3
votes
0answers
59 views

Applied differential equation regarding water clocks

We have a water clock, the shape defined by $r=f(h)$, and the time marks on this water clock are equally spaced. We have to find f(h), and graph $h$ as a function of $r$, assuming the hole through ...
3
votes
0answers
58 views

Writing ODE system with a complex variable

I'm looking at the system of ODEs: $$\begin{cases}\dot{x} = -y + kx + xy^2\\ \dot{y} = x + ky - x^2\end{cases}$$ I'm trying to introduce a complex variable $z = x+iy$ to write this as a single first ...
3
votes
0answers
81 views

Solution for an ODE given only at discrete points

The problem I have: For each $n \in \mathbb N$ I have $$\begin{align} x_0^n & \in \mathbb R \\ h_n & \in \mathbb R \\ x_k^n & = x_0^n + k \cdot h_n \text{ for } k \in \{0,1,\ldots n\} \\ ...
3
votes
0answers
71 views

Voltera equation

Consider the Voltera integral equation: $$ψ(x)=e^{-x}\cos(x)-\int_{0}^{x}e^{-(x-t)}\cos(x)ψ(t)dt$$ How can I solve this equation by converting it to a differential equation? The solution is ...
3
votes
0answers
40 views

A question about solving the nonlinear differential equation $\dot{x} = x(1-x)$

I am aware of the standard solution that makes use of partial fractions. However, I made the following manipulations, in order to be more rigorous with splitting up the differentials before ...
3
votes
0answers
50 views

Arnold ODE Problem

Problem 1 of Section 1.2.4 of Arnold's ODE book asks, "Can the integral curves of a smooth (continuously differentiable) equation $\frac{dx}{dt} = v(x)$ approach each other faster than exponentially ...
3
votes
0answers
44 views

Sources on Jacobi Elliptic Functions

I'm interested in learning more about the Jacobi Elliptic Functions and the associated theta functions. For instance, what was the initial motivation for defining them? What are some applications? Is ...
3
votes
0answers
48 views

Solution of inhomogenous ODE (4th order)

Hello stackexchangers, I have an inhomogenous ODE in 4th order. This ODE is the constitutive law to describe a material by using the "Wiechert model" (p. 15) which is given by $p_0\sigma + ...
3
votes
0answers
46 views

Kinematics of gravity in a non uniform field

I am a first year physics student. I am trying to figure out how to compute position in terms of time for an object falling through non uniform gravity towards the earth, and by extension towards any ...
3
votes
0answers
37 views

the exact graph of the general solution for $x'=\begin{bmatrix} 1 & 1\\ 4& 1 \end{bmatrix}x$

i need someone to give me exact graph of the general solution for $$x'=\begin{bmatrix} 1 & 1\\ 4& 1 \end{bmatrix}x$$ i solved it manually , the general solution is like this ...