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

learn more… | top users | synonyms (1)

0
votes
0answers
5 views

Properties of trajectories generated by subgradient dynamical system

Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a convex function and $x_0\in\mathbb{R}^n$. Consider the subgradient dynamical system: $$ (*) \begin{cases} \dot{x}(t)\in-\partial f(x(t)), \quad \text{a.e....
4
votes
2answers
103 views

Method of Variation of Parameters - Assigning zero works?

I have yet to find a decent answer on this, and so I don't think this question is inappropriate. Also, this question is mainly meant for people that are very familiar with this method. In the method ...
1
vote
0answers
9 views

Picard's Iteration — direct proof of convergence of sequence of integrals to $\tan x$.

The extra credit question on one of the midterm exams for the online version of MIT's Honor's Differential Equations asks us to directly solve the IVP $y' = 1 + y^2$, $y(0) = 0$ and then show that the ...
2
votes
2answers
33 views

Nonuniqueness of Stochastic Differential Equation

Let $B_t$ be the standard Brownian motion, $\mu(t,x)$ and $\sigma(t,x)\ne 0$ are real valued continuous functions where $|\mu(t,x)|+|\sigma(t,x)|$ is NOT Lipschitz continuous, and $$dX_t = \mu(t,X(t)...
1
vote
0answers
51 views

How might an applied mathematician view $ 1/x$, $\ln x$, and $e^x$?

I understand that the natural logarithm was developed by Gregoire de Saint-Vincent and Alphonse Antonio de Sarasa as to represent the area under the curve of the hyperbola $\frac1x$ before the ...
0
votes
3answers
44 views

Solving $(x^2-1)\ddot y-2x\dot y +2y=1$

Solving $$(x^2-1)\ddot y-2x\dot y +2y=1$$ I've solved the homogenous equation: $$y=A(x^2+1)+Bx$$ Where A and B are constants of integration, but I can't for the life of me seem to remember how to ...
1
vote
0answers
16 views

Prove of homotopy equivalence using differential equation

I have to prove that $R^2 \setminus L_1,...,L_n $, where $L_1,..,L_n$ are non intersecting lines, is homotopy equivalent to a wedge sum of $n$ circles. So once I've managed to show that it is possible ...
1
vote
0answers
16 views

Dividing an ODE by a factor that might be zero

The following ordinary differential equation for $F(y)$ is given. $$ (\alpha + \beta A y)\left(\frac{\partial^2 F}{\partial y^2} + \beta^2 F\right) = 0 , $$ The boundary conditions are $F(y \to ...
2
votes
1answer
336 views

A differential equation (nonlinear First-Order)

how to solve this equation: $(Px-y)(Py+x)=h^2P$ that $P=\frac{dy}{dx}$ and $h$ is a constant.
1
vote
0answers
8 views

Strogatz Exercise 6.1.14: how to approximate stable manifold of a saddle point with a series

I'm working through Strogatz's Nonlinear Dynamics and Chaos and am stuck on assignment 6.1.14. We have the following system: $$ \dot{x} = x + e^{-y}$$ $$ \dot{y} = -y $$ which has one fixed point, a ...
0
votes
1answer
17 views

Trying to find a formula in a pricing sheet

I'm trying to find the formula to calculate the price of a product when a certain hight & width are given. I have an excel sheet with what the answers should be. I tried: ...
-1
votes
1answer
23 views

Solving$ x(x-1)\ddot y-x\dot y+y=x(x-1)^2$ by using integrating factor

Solving $x(x-1)\ddot y-x\dot y+y=x(x-1)^2$ given the solution $y_1=x$ I set $y=vx$, have plugged and rearranged. I set my integrating factor to: $$e^\left({\int\frac{x-2}{x(x-1)}dx}\right)$$ ...
0
votes
0answers
15 views

Conservation of charge in the Schrödinger equation

Let us have the Schrödinger equation $iu_t+\Delta u+|u|u=0$, where $u$ is a function, which decays rapidly at infinity. I would like to derive the conservation of charge property, i.e. $||u||_{L^2}=...
1
vote
0answers
21 views

Boundary value problems: eigenvalue and eigenfunction

I'm having trouble in understanding eigenvalues and eigenfunctions in BvP the problem is: $y''$ + $\lambda$$y$ = $0$ $y(0)=0$ $y(2\pi)$ = $0$. Make characteristic polynomial $r^2 + \...
0
votes
0answers
11 views

Exercise at Differential Equations: boundary value problems

$ Let\quad U\quad be \quad a \quad smooth \quad solution \quad of \quad the \quad following\quad boundary \quad value\quad problem: $ $ -cU'+ (F(U))'=εU''\qquad U(-\infty)= A \quad and \quad U(+\...
0
votes
0answers
18 views

integrate equation

I am trying to integrate this equation, however I am not sure which method would be best. $\frac{\dot{a}}{a} = -2 \alpha \frac{\dot{M_1}}{M_1 + M_2}$ All the variables $a, M_1, M_2$ are time ...
0
votes
0answers
15 views

Explaination of this particular approach to form differential equation.

Consider we need to form differential equation of the following: $Ax^2 + By^2 = 1$ where A & B are constants. Now one approach I know of is two differentiate this equation 2 times (since 2 ...
0
votes
0answers
16 views

How to solve complex-valued , inhomogeneous second order differential equations?

I am trying to find a general method to solve the following complex-valued , inhomogeneous second order differential equation $$ -a(x) u''(x)+b(x)u'(x)+c(x)u(x)+d(x)\bar{u}(x)=f(x),\quad x\in\mathbb{...
1
vote
0answers
45 views

How to start an eigenvalue problem

I am stuck on this problem : This is an eigenvalue problem $$\phi''+ \lambda^2 x(x+2)^2 \phi =0\\\phi(1)=0\\ \phi(0)=0$$ I forget this kind of problems... please give me a hint or a clue ,cause I ...
0
votes
1answer
695 views

diffusion equation plot (matlab or maple)

The advection diffusion equation is the partial differential equation $$\frac{\partial C}{\partial t} = D\frac{\partial^2 C}{\partial x^2} - v \frac{\partial C}{\partial x}$$ with the boundary ...
2
votes
0answers
26 views

Show that there aren't negative eigenvalues.

I've been trying to solve this Sturm-Liouville theory problem. Show that the problem: $$\left\{\begin{matrix} y''+(x+\lambda)y = 0\\ y(0)=0\\y(1)=0\end{matrix}\right.$$ doesn't have ...
-1
votes
1answer
30 views

Differential equation function. Cant crack this one up. Tomorrow test

The equation is. $\cos(x) \cdot y'+\sin(x) \cdot y = 2(\cos(x))^3 \cdot \sin(x)-1$ a) Find all the solutions of the associate homogeneous equation. Let $S_h$(homogeneous) denote such a set of ...
1
vote
1answer
22 views

Solve Kolmogorov differential equations for birth-death process with constant rates

I need to solve the Kolmogorov forward equations for a birth-death process whose birth/death rates $\lambda_k,k=0,\ldots$ and $\mu_k,k=1,\ldots $ are constant, i.e., $\lambda_k=\lambda$ and $\mu_k=\mu$...
0
votes
0answers
13 views

Airy's Equation For Series Solutions to Second Order Linear Equations

I am trying to do an example in the textbook for series solutions to second order linear equations, and I am not quite understanding the recurrence relation as follows: I understand and follow the ...
0
votes
1answer
59 views

Solutions to the differential equation $x(x+1)yy' = xy + 1$

I am having trouble solving the linear equation $x(x+1)yy' - xy - 1 = 0$ I will list the steps I followed: (I'm sure I have made some huge mistake.) Divide by $x(x+1)$ $yy' - y/(x+1) - 1/x(...
0
votes
0answers
30 views

Separation of variables and h(y) = 0

Consider the following differential equation $$ y' = g(t) h(y) \tag{1} $$ Solving this by separation of variables, we consider the following cases $ h(y) \neq 0\ \forall\ t$, so we have $ \frac{1}...
0
votes
0answers
26 views

How to find a suitable function for Dulac's criteria in this example?

I have a system of odes $\dot{\mathbf{x}} = \mathbf{f(x)}$ where $\mathbf{x} \in \mathbb{R}^{2}$ and $\mathbf{f(x)}$ is defined below: $$\dot{x} = x- y - x^{3}, \qquad \dot{y} = x+y-y^{3}$$ I would ...
0
votes
0answers
17 views

Direction of a curve given by ODE

Let $a,b \in C(\mathbb{R^2})$ be bounded and $(x_0,y_0) \in \partial B_1(0)$. Consider the ODE system $$ \begin{cases} x'(t)=a(x,y) \\ y'(t)=b(x,y) \\ x(0)=x_0 \quad y(0)=y_0 \end{cases} $$ We know ...
1
vote
0answers
51 views

Finding a parametrization of the solutions of $\frac{dx}{dt}=\frac{\sinh y}{\cosh y+A\cos x}$, $\frac{dy}{dt}=\frac{A\sin x}{\cosh y+A\cos x}$

I am trying desperately to find a parametrization for the following: $\frac{dx}{dt}=\frac{\sinh y}{\cosh y+A\cos x}$ $\frac{dy}{dt}=\frac{A\sin x}{\cosh y+A\cos x}$ I tried to devide the equation ...
1
vote
0answers
12 views

Equation of silhouette from an arbitrary viewpoint

A two parameter $(u,v)$ surface in $\mathbb R^3$ when viewed from a point at infinite distance casts a shadow on any given plane. What ODE/PDE describes its envelope of its silhouetted projection? ...
0
votes
1answer
17 views

“Hessian” differential equation

In my homework, I'm given the following problem: Let $f: \mathbb{R}^n \to \mathbb{R}$ be a twice differentiable function. For an $\alpha \geq 2$, let: $$f(\lambda x) = \lambda^a f(x)$$ for ...
0
votes
1answer
14 views

What is the missing step to solve the two ODE?

I was reading an example of an exercise of variational calculus where they get the system of equations: \begin{equation} F_{y'y'}y''+F_{y'z'}z''=0;\;\;F_{y'z'}y''+F_{z'z'}z''=0, \end{equation} where $...
1
vote
2answers
43 views

Solved ODE by two different methods; unable to prove the solutions are the same.

I have a first order ODE: $$\frac{dy}{dx}+\frac{2y}{x}=\frac{e^x}{x^2}\tag{1}$$ Noting that this is in the form $$\frac{dy}{dx}+P(x)y=Q(x)\tag{2}$$ So an integrating factor method can be used. ...
2
votes
0answers
21 views

Stability of ODE with periodic coefficients / Periodic solutions

Given the ODE system $$ \left\{ \begin{array}{l} \dot x = -2x - z \cos t, \\ \dot y = x \sin t - y, \\ \dot z = -4z + \sin^2 t. \end{array} \right. $$ I am asked to: 1) Examine the stability of ...
1
vote
0answers
24 views

Difficulties understand the series solution of $(1-x^4)y''-8x^3-12x^2y=0$

Solve: $(1-x^4)y''-8x^3-12x^2y=0$ using the solution: $$y=\sum_{n=0}^{\infty}a_nx^n$$ Let's differentiate y: $$y'=\sum_{n=1}^{\infty}(n)a_nx^{n-1}$$ $$y''=\sum_{n=2}^{\infty}(n)(n-1)a_nx^{n-2}$$ ...
1
vote
1answer
51 views

does a linear differential equation have a well defined initial value problem if a term diverges at initial 'time'?

Suppose I have a differential equation like the following: $$\frac{d^2x}{dt^2}+t^2x=0$$ And I've to put initial conditions at $t=-\infty$. Now the $t^2$ bit seems to diverge at $t=-\infty$. Is that ...
2
votes
1answer
31 views

Why does a trajectory take infinite time to reach a critical point?

I have to prove that: For an ODE system $x'=F(x,y),y'=G(x,y)$ where $F,G$ are smooth, any trajectory, which doesn't start at a critical point, cannot reach a critical point in finite time. I have ...
2
votes
1answer
50 views

Clairaut's form of $(x\frac{dy}{dx}-y)(y\frac{dy}{dx}+x)=a^2\frac{dy}{dx}$

Question is to find the Clairaut's form of differential equation $$(x\frac{dy}{dx}-y)(y\frac{dy}{dx}+x)=a^2\frac{dy}{dx}$$ I know clairaut's equation is of the form $y=x\frac{dy}{dx}+f(\frac{dy}{dx})$...
2
votes
0answers
20 views

(Conceptual_Calculus) Differential Conditions v. Derivative Conditions

I have few questions regarding the reason we learn about the differential conditions in higher dimensions and dealing with multivariable calculus. In the context of optimization (e.g. finding ...
3
votes
1answer
289 views

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
votes
1answer
33 views

Determine the solution of the following system of Differential Equations

Determine the solution of the following system: $$\dot{x_{1}}=-x_{1}$$ $$\dot{x_{2}}=-x_{2}+x_{1}^{2}$$ $$\dot{x_{3}}=x_{3}+x_{1}^{2}$$ The first equation clearly has solution $x_{1}(t)=c_{1}e^{-t}$...
2
votes
1answer
15 views

Can someone explain where this comes from for the cauchy-euler equation in the case of double roots

I am not sure how the following was derived in my textbook given by I understand that the solution we are seeking is $y = x^r$ and the fact that we obtained the first solution as $y = c_1 x^{r_1}$ ...
2
votes
1answer
962 views

Wrong answer for this differential equation temperature problem.

(a) An object is placed in a 68°F room. Write a differential equation for H, the temperature of the object at time t. ANSWER: dH/dt = -k(68 - H) (b) Give the general solution for the differential ...
3
votes
3answers
42 views

Differential Equation Initial Value Problem

Here is a pretty standard initial value problem that I'm having a little trouble with. $$(\ln(y))^2\frac{\mathrm{d}y}{\mathrm{d}x}=x^2y$$ Given $y(1)=e^2$, find the constant $C$. So I separated and ...
1
vote
2answers
37 views

Differential equation $\left(x^2+xy\right)y'=x\sqrt{x^2-y^2}+xy+y^2$

I am not sure which type of differential equation this falls into: $$\left(x^2+xy\right)y'=x\sqrt{x^2-y^2}+xy+y^2$$ any hints? P.S. I first tried reornazing it so I have $y'$ alone, and hoping that I ...
0
votes
0answers
20 views

System of N-1 first order ODEs

I've to elaborate this system of N-1 first order ODEs for i = 1,2,...,N and n =1,2,...,N. $ (z_3)_n^.(t)= \sum_{i=1}^{N-1} (C_3)_{ni}(z_3)_i(t)+ (z_1)_n(t)\sum_{i=1}^{N-1} (A_3)_{ni}(z_3)_i(t)$ For N=...
-10
votes
0answers
56 views

Need someone to refer through via e-mail for math help? [on hold]

Alright, so I am a third-year mathematics student, and I took a differential equations course last semester, and failed. I found the lectures to be very vague and too problem-specific, so I would go ...
7
votes
4answers
275 views

Verify $y=x^aZ_p\left(bx^c\right)$ is a solution to $y''+\left(\frac{1-2a}{x}\right)y'+\left[(bcx^{c-1})^2+\frac{a^2-p^2c^2}{x^2}\right]y=0$

In order for the question that I have to make any sense I must first include some background information as given in my textbook: The standard form of Bessel's differential equation is $$x^2y^{\...
1
vote
0answers
75 views

Verify $y=x^{1/2}Z_{1/3}\left(2x^{3/2}\right)$ is a solution to $y^{\prime\prime}+9xy=0$

This question is a sequel to this previous question. As before, some background information is needed first as follows from my textbook: The standard form of Bessel's differential equation is $$x^...