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

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Is $x=0$ an ordinary or singular point? Two conflicting textbook solutions that use the same reasoning.

We're asked to determine whether $x=0$ is an ordinary point or singular point for the following two ODEs: $$\begin{align*}x y''+\sin x\,y&=0&(1)\\\\ x y''+(1-\cos ...
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0answers
17 views

Show that this function is differentiable at all points [on hold]

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1answer
27 views

Use series methods to find solution corresponding to..

Use series methods to find solution corresponding to $a_0 = 1$ for the equation $(x+1)y' - y = 0$ Here is my work. Can someone verify that I have the correct solution: So for my final solution I ...
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0answers
11 views

Calculus scenario involving instantaneous and speed (sequences)

The scenario is nearly always the same as Wilie is standing at the end of a road that is 1 kilometer long, and there at the other end is that Roadrunner, he’s just standing there, sticking his tongue ...
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2answers
22 views

Real analysis: simple second order ODE

I'm studying real analysis at the moment (just covered the mean value theorem, constancy theorem, applications to DEs etc.) and have run across this question that I'm stuck on. Any help would be much ...
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1answer
26 views

Spectrum of operator $Af(t) = \int_0^{t^2} f(s)ds$ on $L^2[0,1]$

Consider a linear operator $A\colon L^2[0,1]\rightarrow L^2[0,1]$ that acts as follows: $$Af(t) = \int_0^{t^2} f(s)ds$$ The problem is to compute its spectrum. I know that the operator is compact ...
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2answers
50 views

Why is this the eigenvector?

For the eigenvector how are they getting \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} when you have \begin{bmatrix} 0 & -1 & -1 \\ 0 & -1 & -3 \\ 0 & 0 & -2 \end{bmatrix} ...
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1answer
12 views

Need help solving a system of iterative differential equations

Here I have a system of differential equations: $u_{0}''=-1$ $u_{0}u_{0}''+u_{1}''=-1$ $u_{2}''+u_{1}''u_{0}+u_{0}''u_{1}=-1$ $u_{3}''+u_{2}''u_{0}+u_{1}''u_{1}+u_{2}u_{0}''=-1$ ...
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2answers
42 views

Solve $2y^{(5)}-7y^{(4)}+12y'''+8y''=0$ [on hold]

Find the general solution of higher order linear differential equation? Find the general solution of Differential equation using auxiliary equation? $$2y^{(5)}-7y^{(4)}+12y'''+8y''=0$$
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1answer
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Initial value problem through origin

$\frac{dz}{dt}=8t*e^z$, Through the origin I have never done an initial value problem before, but I took it to mean that it gave me the initial value of the differential equation (0, 0) and that I ...
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20 views

Asymptotic Behavior of Differential Equation

physicist here. I'm studying some problems that involve the use of differential equations. The professor of the course has indicated that usually variable changes used to simplify the equations come ...
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1answer
21 views

two set of ordinary differential equations

Can you please check my calculation below. Thanks
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1answer
72 views

What is the general skill to solve third order ordinary differential equation? [on hold]

What is the general skill to solve third order ordinary differential equation, and just list the references? Those are with or without trigonometric, logarithms, exponential and with the typical x ...
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1answer
19 views

Solving system of ODEs using different methods

So here I have my system of ODEs with its initial conditions: $y_{0}''+1=0$ $y_{1}''+y_{0}'+2y_{0}'y_{1}'=0$ $y_{2}''+2y_{1}'^2=0$ The initial conditions are $y_{0}(0)=1$ and ...
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2answers
23 views

Solving Bernoulli equation transformation

I'm trying to solve the Bernoulli's equation via perturbation method but I need some help understanding how its done: We start off with $y'=-y+\epsilon y^2$ with $y(0)=1$. Then how is it possible ...
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PDE Heat Equation with Variable Coefficient {Second ODE Variable Coefficient}

Another PDE question: If I have a non constant coefficients in my heat equation (PDE), how do I solve it? For example we have: $\frac {\partial T}{\partial t} =\frac {\partial ^2 T}{\partial r^2} + ...
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3answers
28 views

Find all line equations that are tangent to $x^3 - x$ and pass through $(-2,2)$

So I have the equation: $f(x) = x^3 - x$ So we know that the slope of the curve for some $x$ is given by: $f'(x) = 3x^2 - 1$ And need to find equations of lines that are tangent to that curve, ...
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0answers
11 views

How to apply the laplace transform to this second order ODE?

Can I apply the Laplace transform on a the following second order nonlinear PDE? $$ \frac{\partial y}{\partial t}=\frac{\partial^2 y^n}{\partial x^2}$$ where $n$ is a natural number? I mean apply the ...
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1answer
19 views

Existence of nonconstant periodic solution

Show that the given system has a nonconstant periodic solution: $$\frac{dx}{dt}= 8x - 2y - 4x^3 - 2xy^2$$ $$\frac{dy}{dt}= x + 4y - 2y^3 -3x^2y$$ Above is my question. I tried to use the Poincare ...
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1answer
10 views

Palais–Smale compactness condition

Can someone explain the essence of Palais–Smale compactness condition used in the Mountain Pass Theorem, in particular its weak formulation?
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5 views

$\Gamma$-convergence (Gamma-convergence) and PDEs?

My question is about the applying calculus of variations to solving Partial Differential Equations. In particular, what is the idea behind using $\Gamma$-convergence to find weak solutions of PDEs? ...
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1answer
17 views

Writing down solutions of differential equations [on hold]

Say the solution to a differential equation is $C_1f(t)+c_2 i g(t)$. We can write this as $a_1f(t)+a_2g(t)$, where $a_2=c_2i$? Or do the coefficients have to be real numbers?
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Fourier Expansion of Hill's lunar problem

all! For my class I have to expand the following equation $y''(x)=4(\omega^2+q(x))y(x)$ in Fourier coefficients $y(x)=\frac{1}{2}y_0 + \sum^\infty_{n=1}y_n \cos(2nx)$ $q(x)=2\sum^\infty_{n=1}t_n ...
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2answers
676 views

Is it mathematically valid to separate variables in a differential equation?

I read the following statement in a book on Calculus, as part of my mathematics course: Technically this separation of $\frac{dy}{dx}$ is not mathematically valid. However, the resulting ...
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27 views

Finding normalized eigenfunctions for $y'' + \lambda y = 0$

Find the normalized eignefunctions for $$y'' + \lambda y = 0$$ $$y(0)=0, y(\pi)-2y'(\pi)=0.$$ My teacher gives me this hints: Consider$$(py')'+qy+\lambda ry=0$$ where $p, p', q, r$ are ...
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0answers
25 views

Help with a Differential Equations problem. [on hold]

Having problems with this one, i just got into Diferentials Equations.
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0answers
8 views

Given a piecewise initial condition, how can the characteristic curve x be sketched when the solution x does not contain u terms?

The charac equation for x: $$\frac{\text{dx}}{\text{d$\tau $}}\text{=2t}$$ The solution x is $$x=t^2+x_0$$ Note that $$\tau=t$$ There is a problem. In order to sketch x, I require some ...
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1answer
34 views

Solving a system of coupled differential equations [on hold]

The system is given by: \begin{align} 2x''&=-6x+2y \\ y''&=2x - 2y + 40\sin(3t) \end{align} The textbook did not go more deeply to give the solving technique of these type of problem instead ...
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2answers
33 views

Is $L=\sin^2(t) \frac{d}{dt}$ a linear differential operator?

Consider the differential operator $$L=\sin^2(t) \frac{d}{dt}$$ If it acts on the sum of two functions, $y_1(t)$ and $y_2(t)$, you get $$\begin{align*} L(y_1(t)+y_2(t))&=\sin^2(t) ...
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1answer
22 views

Solving ODES in PDE

The PDE given as: $$t^2u_t-\text{yu}_x+\text{xu}_y\text{=0}$$ The characteristic equations are: $$\frac{\text{dt}}{\text{dt}}=t^2$$ $$\frac{\text{dx}}{\text{dt}}\text{=-y}$$ ...
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1answer
30 views

Manipulating series to find the recursive formula

Ok so I am stuck. I need to get all the $n$'s to $=0$ but I can't reduce my series which has $n=2$ to $0$ because then I will have undone all my work in the first place to get all the $X^n$'s to the ...
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1answer
20 views

How do I express this system of differential equations in polar coordinates?

I'm supposed to express this system of differential equations in polar coordinates. $\begin{cases} \frac{dx}{dt}=\mu x-\omega y-x(x^2+y^2)\\\frac{dy}{dt}=\omega x+\mu y-y(x^2+y^2)\end{cases}$. I'm a ...
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2answers
14 views

Third order non-homogeneous differential equation

I have no idea on how to work this out. I've tried variation of parameters, undetermined coefficients, making it into a system, etc. $$y'''+2y''+5y'+20e^{-x}\cos(2x)=0$$
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1answer
18 views

Index of differential function

Is it valid to say: $$\frac{d}{dy} \left( \frac{du(y)}{dy} \right)^n = \left(\frac{du(y)}{dy}\right)^{n+1}$$ If so, why?
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What method do we use to find the solution?

Find the solution of the initial and boundary value problem $$u_t(x,t)-u_{xx}(x,t)=0, x>0, t>0, \\ u(x,0)=f(x), x>0,\\ u(0,t)=0, t>0 $$ (The solution should be expressed as an integral ...
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0answers
33 views

What is the exact solution to this PDE?

I'm in a numerical methods class for my senior year of college, and it's been about 3 years since I took diff eq. We have a problem in which we are using numerical methods to approximate the solution ...
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1answer
27 views

Three-Variable Differential Equation Stability

Discuss the stability of the equilibrium points $(1,0,0)$ and $(1,1,0)$ for the system: \begin{align} x' &= y - y^2\\ y' &= z\\ z' &= x - \cos{z} \end{align} I have found the ...
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1answer
25 views

How to describe behavior of population system, given by system of ODEs?

So I have a system of equations:$$x'(t)=x(t)(4-2x(t)-y(t))\\y'(t)=y(t)(3-x(t)-y(t)) $$ What I understand so far is: if we have x being the population of prey and y is the population of predators. x ...
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0answers
14 views

Solution space of Linear homogeneous differential equation

The solution space of a L.H.D.E of order n is a vector space spanned by n base vectors, right? So any solution is then a vector of the solution space -> a linear combination of the base vectors. But ...
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58 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 ...
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What does affine invariance mean in the context of the Newton's method?

The textbook Numerical Solution of Boundary Value Problems for Ordinary Differential Equations (by Ascher, Mattheij, and Russell) states on page 329: [W]e observe that Newton's method is affine ...
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1answer
42 views

Wave Equation Partial Differential EEquation

Basically I got a simple wave equation with an extra twist. The PDE is $\frac {\partial^2 y}{\partial t^2} = c^2\frac {\partial^2 y}{\partial x^2} + L $ with homogeneous boundary condition As usual, ...
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1answer
29 views

An application of Implicit Function Theorem in differential equations?

Let $f$ be a continuous function from $\Bbb R^3 \to \Bbb R$. By a solution of the differential equation $$f(x,y,\dot{y}) = 0$$ We mean a function $y\colon U \subset \Bbb R \to \Bbb R$ where $u$ is an ...
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2answers
52 views

differential equation question $\frac{dy}{dx} = \frac{2xy}{x^2 + y^2}$ [on hold]

how do you solve this ? $$\frac{dy}{dx} = \frac{2xy}{x^2 + y^2}$$ thank you in advance!
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0answers
18 views

Show that the solution of the Cauchy problem $x(t,t_0,x_0)$, $x(t_0)=x_0$ is definite for all $t\geq t_0$. [on hold]

Consider the system: $$x' = A(t)x + b(t)$$ where matrices $A(t)$ and $b(t)$ are only integrable on compact sets of $\mathbb{R}$. Show that the solution of the Cauchy problem $x(t,t_0,x_0)$ is ...
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the Fisher equation has no positive traveling wave solution [on hold]

Use the linearization method to prove that for any $c\in(0,2)$,the Fisher equation$u_t=u_{xx}+u(1-u)$has no positive traveling wave solution $U(x+ct)$ with $U(-\infty)=0$
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0answers
18 views

Use the persistence theory to find a set of sufficient conditions for two species competitive ODE system [on hold]

Use the persistence theory to find a set of sufficient conditions for two species competitive ODE system $$\frac{du_1}{dt}=u_1(b_1-a_{11}u_1-a_{12}u_2)$$ ...
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2answers
65 views

Can somebody please show me the necessary steps to solve this Calculus problem?

I have a homework assignment that asks me to solve the differential equations and it gives me: \begin{align*} xy^2y' & = 2-x\\ y''+4y & = 8x\\ y(1)& =1 \end{align*} Are these three ...
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2answers
36 views

Analytic solution to Poisson equation

I need to find the analytic solution to this equation, in order to compare it with solution I get from using a numerical solution. However, I have not been able to find the solution. I think I can't ...
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1answer
10 views

Central Difference Method

Solve the following using the central difference method: $y(x)= y'+ y + 2x$ where $0 < x < 4$ with $n=4$ subintervals (thus $h=1$). Given that $y(0)=0$ and $y(3)=1$, find $y(1)$. Really ...