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

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3
votes
2answers
46 views

Continuous dependence of maximal interval on right hand side for ode

Let $f,f_n\colon\mathbb{R}\times(0,1)\to\mathbb{R}$ be local Lipschitz continuous functions and assume they form the right-hand-side of some ordinary differential equation's $\dot{x} = f(t,x)$ and $\...
3
votes
1answer
52 views

The meaning of dependent and independent variables in ODEs

This thought has occured to me a few days ago, and now I am puzzled about some fundamental properties /definitions in the theory of differential equations. Suppose I have an ode $$\frac{dy}{dx}=y^{2}$...
2
votes
0answers
49 views

Why is this exponential bounded?

Let $A$ be a square real $n\times n$ matrix whose eigenvalues have real part $\leq 0$. Let $\lambda_1,...,\lambda_l$ be eigenvalues of $A$ whose real part is zero. Suppose that for each $1\leq j \leq ...
0
votes
1answer
75 views

Suppose that a room containing 1800 cubic feet of air is originally free of carbon monoxide (CO).

Suppose that a room containing 1800 cubic feet of air is originally free of carbon monoxide (CO). Beginning at time $t=0$, cigarette smoke containing 5% CO is introduced into the room at a rate of $0....
0
votes
1answer
23 views

A basic differential inequality (in the proof of local uniqueness of ODE via fixed point method)

Hi I am having trouble getting a differential inequality of the like $$\frac{d}{dt}|y|^2\le 2K|y|^2$$ while reading the chapter on uniqueness of ODE. Could anyone explain the first line of equation (...
0
votes
0answers
18 views

Boundary value problem differential equation for a linear 2nd order ODE

I'd like to solve this BVP: $$-k\frac{d^2T}{dz}+pCv\frac{dT}{dz}+\frac{2h}{R}(T-T_{out})=0$$ I'm having trouble with the particular solution and how to evaluate the constants. Boundary values: $$T(...
1
vote
1answer
14 views

Piecewise differential equation continuity

My question doesn't have anything to do with solving the below problem, but I have listed it below for convinience. The question states we have to solve $y''+4y=g(x)$ with $g(x) = \begin{cases} ...
1
vote
0answers
44 views

A tank contains 100 kg of salt and 2000 L of water.

A tank contains $100\;\text{kg}$ of salt and $2000\;\text{L}$ of water. Pure water enters a tank at the rate $10 \;\text{L/min}$. The solution is mixed and drains from the tank at the rate $13 \;\text{...
2
votes
0answers
37 views

Drawing Trajectories in a State Space (is Energy Conserved?)

There's something I'm slightly confused about regarding drawing the trajectory of a particle in state space $(x,v)$, where $v:=x'$. Here, I'm only working with $x\in\mathbb{R}$. Suppose a particle ...
0
votes
1answer
52 views

Green's function For Helmholtz Equation in 1 Dimension

We seek to find $g(x)$ with $x\in \mathbb{R}$ that satisfies $$(k^2 + \partial_x^2)g(x) = \delta(x)$$ Based on what I obtain in reference material, we try the following $$q(x) = e^{-ik|x|}$$ $$\...
0
votes
1answer
107 views

Suppose you deposit 8500 dollars into a savings account earning 5 percent annual interest compounded continuously

Suppose you deposit 8500 dollars into a savings account earning 5 percent annual interest compounded continuously. To pay for all your music downloads, each year you withdraw $900 in a continuous way. ...
0
votes
0answers
31 views

How do I find the particular Integral for a PDE?

How do I solve $(D^2 - 2DD')z = x^3y + \Bbb e ^{5x}$? The auxiliary equation is $m^2 – 2m = 0$. Solving it, I get $m = 0$ and $m=2$. Hence the complementary function is $f_1 (y) + f_2 (y+2x)$. How ...
1
vote
1answer
45 views

Question about phase portrait and invariant subspaces

I am trying to understand why the eigenvectors are the $A$ invariant subspaces of a phase portrait. An A-invariant subspace is defined by the relation $AV \subseteq V$ where $V$ is a subspace and $A$ ...
0
votes
1answer
28 views

Help evaluating this limit

Evaluate the limit $$\lim_{\alpha\to\omega}-\frac{\alpha r_0}{\omega(\omega^{2}-\alpha^{2})}\sin(\omega t)+\frac{r_0}{\omega^{2}-\alpha^{2}}\sin(\alpha t)=\frac{r_0}{2\omega^{2}}sin(\omega t)+\frac{...
0
votes
4answers
25 views

Particular solution to this differential equation

Can someone help me find the particular solution to this diff equation: $\frac{d^2y}{dx^2}-a\frac{dy}{dx}=C$
0
votes
1answer
27 views

Solving differential equation and obtain expressions with unknowns?

I have the following differential equation $my'' + \beta y' + mg = 0$ , with condition $y(0)=0$. I need to solve the equation and obtain expressions for $y(t)$ and $y'(t)$. I have attempted to use ...
0
votes
0answers
37 views

Total differentiation equation concerning homogeneous function

I was asked to prove the following statement: Suppose $P(x,y),Q(x,y)$ are both $n$-th homogeneous functions ($n\ne-1$), i.e., $$P(tx,ty)=t^nP(x,y),\, Q(tx,ty)=t^nQ(x,y)\quad\forall t\in\Bbb R$$ ...
1
vote
1answer
77 views

Help in Solving a linear Partial differential equation

I can not to solve the following equation $$(*) \qquad u''(r) +2n\coth(r)\,u'(r)+ (n^2+\lambda^2) \, u(r)= 0 \quad \mbox{with} \, r>0$$ where $n\in \mathbb N$ and $\lambda \in \mathbb C $. That I ...
0
votes
1answer
24 views

Find the differential equation

What is the differential equation for $ y = Ln(cos(x - c_{1})) + c_{2} \ , \quad c_{1} , c_{2} \in \mathbb{R} $ $ y^{\prime} = -tan(x-c_{1}) $ , and hence $ y^{\prime \prime}= - (1+tan^{2}(x-c_{1})) ...
1
vote
1answer
46 views

ODE Hyperbolic fixed point study

Here is a problem I can not solve : Let $X(x, y)$ be a $C^{\infty}$ vector field defined on $\mathbb{R}^2$ such that $X(x, y) = (y + a(x, y), x+ b(x, y))$ where $a, b : \mathbb{R}^2 \to \mathbb{R}$ ...
1
vote
1answer
21 views

Differential Equation (First order with separable variable)

Given that $\frac{dy}{dx}=xy^2$. Find the general solution of the differential equation. My attempt, $\frac{\frac{dy}{dx}}{y^2}=x$ $\int \frac{\frac{dy}{dx}}{y^2} dx=\int x dx $ $-\frac{1}{y}=\...
0
votes
2answers
54 views

ODE time of life of a solution

Consider following the ODE : $$x'=1-t^2+x^2$$ What is the time of life of a solution from this equation ? The Cauchy-Lipschitz theorem proves that a solution for the Cauchy boundary $x(0)=x_0$ lives ...
-2
votes
3answers
53 views

How to solve $(D^2+4)y=\sin x$?

find special solution of $(D^2+4)y=\sin x$ $$y_c=c_1\sin 2x+c_2\cos2x$$ $$y_p=v_1\sin2x+v_2\cos2x$$ $v_1'\sin2x+v_2'\cos2x=0 \tag 1$ $2v_1'\cos2x-2v_2'\sin2x=\sin x\tag 2$ I tried to solve these ...
0
votes
0answers
32 views

Second Order Differential Equation Solubility

I have the second order differential equation $\frac{\mathrm d^2x}{\mathrm d t^2}+\omega^2x=\omega^2l-g\cos(\omega t)\frac{l-x}{\sqrt{x(2l-x)}}$ Were it not for that term at the end, things would be ...
1
vote
1answer
32 views

Integrating $ \frac{\mathrm{d}^{2}v}{\mathrm{d}y^{2}} = \frac{\mathrm{d}p}{\mathrm{d}x} $

How would I go about integrating this to figure out what $\mathrm{d}v/\mathrm{d}y$ is? The bounds on $y$ is ($H$--> constant upper, and $y$ varied lower). I know how to generally do it but I'm not ...
2
votes
2answers
52 views

A simple question about the solution to $mx''=-kx$, mass on a spring

We did this in class, and we found the solution to $m\ddot x=-kx$. I am fine with the whole solution, i.e.: -let $\omega ^2= k/m$ to turn the problem into $\ddot x+\omega ^2x=0$ ...(1); -Guess that ...
1
vote
0answers
27 views

Variable separable differential equation

Without several combination trials is there an algorithm during check for solution whether a differential equation has its variables amenable to being separated or not ?
0
votes
5answers
89 views

Given 1 solution to a differential equation, find another solution

How would I find a linearly independent solution to a differential equation given a solution to the differential equation. For example, I have this question: One solution of Hermite's ...
0
votes
1answer
29 views

Solution of the IVP $\frac{\partial{u}}{\partial{t}}+\frac{\partial^2{u}}{\partial{x^2}}=0$.

Given $\frac{\partial{u}}{\partial{t}}+\frac{\partial^2{u}}{\partial{x^2}}=0$ with $u(x,0)=\sin{\pi x}$. I don't know how to derive a solution of it. But I just want a hint to move forward. Not a ...
0
votes
2answers
35 views

What is $\frac{d}{dt}xcos(y)$?

$\frac{d}{dt}x = \dot{x}$ and $\frac{d}{dt}y = \dot{y}$ but how do I differentiate something like this with respect to time? $\frac{d}{dt}xcos(y) = ?$
0
votes
1answer
36 views

No eigenvalues for ODE in semi-infinite domain?

I cannot understand the existence of eigenvalues of ODE regarding semi-infinite domain. Before post the question, what I've already known is as follows. Given a linear 1D wave equation as follows $$ ...
0
votes
2answers
43 views

Solving $\frac{dx}{dt}=-2x-2y, \frac{dy}{dt}=-2x+y$ with initial condition $(x(0), y(0)) = (1, 0)$

The course I am taking is using the text Differential Equations by Blanchard, Devaney, and Hall, and I want to solve the following problem: Solve $$\frac{dx}{dt}=-2x-2y, \quad \frac{dy}{dt}=-2x+y$$...
-2
votes
1answer
33 views

Differential equation : $\frac{dy}{dx} -\frac{y}{x}=2x^2, x > 0$

Problem : Solve the differential equation : $$\frac{dy}{dx} -\frac{y}{x}=2x^2, \quad x > 0$$ Solution : $$\frac{dy}{dx} + \left(-\frac{1}{x}\right)y =2x^2 \tag i$$ It is linear ...
6
votes
2answers
279 views

Why is it wrong to derive the chain rule this way?

My book says that the chain rule can stated as $$\dfrac{dy}{dx} = \dfrac{dy}{dt} \dfrac{dt}{dx}$$ However, it the book says that it is incorrect to reason that the chain rule is true because the $dt$'...
1
vote
1answer
30 views

Inhomogeneous Boundary Value Problem

I am trying to solve the following BVP $a V^{\prime \prime}(x) + b[c - x]V^{\prime}(x) + x V(x) = 0$ with the boundary conditions $V(-\infty) = 0 \quad \mbox{and} \quad V(0) = 1$ I tried some ...
0
votes
1answer
14 views

Invariance of radiative transfer equation in the absence of absorption and emission

The question is asking me to show that the line-of-sight intensity of radiation is invariant when there is no emission or absorption. Starting with the radiative transfer equation: $$ \frac{d}{dz}I_{...
1
vote
0answers
45 views

Finding the spectral representation of the Delta function given the Green function of an operator

I'm working through the problems of a book on Sturm-Liouville problems. In a problem I found the Green function for the following SLP2 problem $$\frac{-d^2g}{dx^2}-\lambda g=\delta(x-\xi)$$ $$g(0,\...
0
votes
0answers
38 views

Terminating numerical integration on limit cycle to compare ODE solvers

I have (hand-)written two basic ODE solvers in Matlab for a numerical analysis course, and need to compare their effectiveness by numerically integrating an autonomous nonlinear first-order system in ...
0
votes
1answer
18 views

Inhomogeneous Bessel function of integer order

I'm looking for a specific solution of: $$z^2 w'' + z w' + (z^2 - \nu^2) w = a^2$$ where a is a constant (with a = 0 this would be the standard form for the integer order Bessel equation). The a = ...
6
votes
0answers
147 views

Transforming the solutions of $\dot x = f(\mathbf{x}, t)$ and $\dot x = B(\mathbf{x}, t)f(\mathbf{x}, t)$ into each other

How can I prove the following theorem? If the function $B(\mathbf{x}, t)$ is strictly positive, then the solutions of the two differential equations $\dot x = f(\mathbf{x}, t)$ and $\dot x = B(\...
0
votes
0answers
24 views

Fourier transform to determine stability of fixpoint of equation with temporal convolution

Given the differential equation \begin{align} \frac{d v}{d t} = - v(t) + \kappa * v \end{align} where $\kappa$ is some linear temporal filter (like a sum of two exponentials, for instance) and $...
1
vote
3answers
58 views

Solving the second-order, linear, inhomogeneous ODE $y'' - 2y'\tan(x)-y=\sin(x)$

I have the following ODE: $$y'' - 2y'\tan(x)-y=\sin(x)$$ I am at a loss where to start. All the methods described in my textbook assume knowledge of the complementary function to solve $2$nd ...
0
votes
3answers
55 views

Solve $x^2 \frac{d^2y}{dx^2}+4x \frac{dy}{dx} + 2y= \cos x$

Solve $$x^2 \frac{d^2y}{dx^2}+4x \frac{dy}{dx} + 2y= \cos x$$ My attempt: Let $x=e^u$, and $D=\frac{d}{du}$ then the given equation becomes $$(D+2)(D+1)y=\cos(e^u)$$ Solving the corresponding ...
0
votes
0answers
17 views

How to estimate $\displaystyle\widehat{u}(t, \xi)=\frac{1}{\mu(t, \xi)}\int \mu(t, \xi) \widehat{f}(t, \xi)\ dt$?

Consider the ordinary differential equation: $$\widehat{f}(t, \xi)=D_t \widehat{u}(t, \xi)+c(t)p(\xi) \widehat{u}(t, \xi),$$ where $f, u\in C^\infty(\mathbb T^{1+n})$, $c\in C^\infty(\mathbb T)$ and $...
1
vote
0answers
36 views

Transform the equation using a substitution

I'm trying to rewrite this equation: so that it's in the diffusion equation form:
3
votes
0answers
81 views

Door mechanism differential equation

I have been wondering about a door mechanism I have seen. It has a wire attached to the upper corner of the door and from there to the corresponding corner in the door frame, where a weight hangs from ...
1
vote
0answers
38 views

For which value of $r$ is the nonlinear dynamical system dissipative?

I could really use some help with the following: When I have input $u$ and output $y$ of the following nonlinear dynamical system, How can I determine for which values of $r$ this system will be ...
0
votes
2answers
31 views

show that the orbit represented by the function r() is an ellipse

let $r(θ)=a(1-β^2)/(1+β\cos\theta)$ representing the distance from the Sun to a planet. With $0<β<1$, show that the orbit represented by this function $r(θ)$ is an ellipse described by $(x+\sqrt{...
0
votes
2answers
20 views

Why is the solution to this ODE as follows?

$rV = \pi x − f + \mu x \frac{\partial V(x)}{\partial x} + 0.5\sigma^2 x^2 \frac{\partial^2 V(x)}{\partial x^2}$ Why is the general solution given by: $V(x) = A_0 + A_1 x + A_2 x^\lambda + A_3 x^\...
0
votes
1answer
29 views

How to complete this partial differential equation?

$T = \frac{1}{2}M_{w}\dot{x}^{2} + \frac{1}{2}I_{w}\frac{\dot{x}^2}{r^2} + \frac{1}{2}M_{b}((\dot{x} + l\dot{\theta}cos(\theta))^2 + (l\dot{\theta}sin(\theta))^2) + \frac{1}{2}I_{b}\dot{\theta}^{2}$ $...