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

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Find the minimizer of the functional

Find the minimizer of the functional $ l= \int u(t) $ with $u(1)=u(1)=0 $ subject to $g=\int $$\sqrt{1+u'(t)} dt $ I want to solve it using E-L equation first $l^*=l- \lambda g$ then i used e-l ...
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0answers
15 views

PDE question: heat equation (third order??)

I am familiar with the usual heat equation, however, my lecturer gave me this problem and it does not look like anything I have ever seen (in my whole entire life and I am not just being dramatic). ...
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0answers
7 views

basic reproduction number of a simple SEIR-model

the normal SEIR-model is: $\begin{array}{rll} \displaystyle{\frac{dS}{dt}}&=\mu N -\mu S -\beta \frac{I}{N} S & \text{Susceptible} \\ \displaystyle{\frac{dE}{dt}}&= \beta \frac{I}{N} S ...
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0answers
21 views

Stability of equilibriums

The question is to find the stability of the equilibriums of the system: $$\frac{dx}{dt}= 8x - 2y - 4x^3 - 2xy^2$$ $$\frac{dy}{dt}= x + 4y - 2y^3 -3x^2y$$ There are 3 equilibriums, $(0,0), (1,1), ...
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1answer
13 views

How to solve Sturm-Liouville problem

Find all the eigenvalues and eigenfunctions of Sturm-Lioville problem: $$y'' + (1 + \lambda)y = 0$$ $$y(0) = y \left(\frac{\pi}{2}\right) = 0$$ Can someone please tell me how to solve this? Because ...
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2answers
27 views

Repeated root case for ay''+by'+cy=0

To solve the ODE $a y''(t)+b y'(t) +c y(t) = 0$, where $a,b,c$ are constant, we solve the characteristic equation $ar^{2}+br+c=0$. In the case when the roots are two repeated roots, i.e,. ...
2
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2answers
29 views

Solve $(x^2 + 1)y'' - 6xy' + 10y =0$ using series method

Use series methods to solve: $(x^2 + 1)y'' - 6xy' + 10y =0$ a) Give the recursion formula b) Give the first two non-zero terms of the solution corresponding to $a_0 = 1$ and $a_1 = 0$ ...
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0answers
15 views

Equillibria to Differential Equations

I am wondering what the exact definition is of an equilibrium to a differential equation. It seems like the general consensus implies that a differential equation will only have an equilibrium if it ...
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0answers
9 views

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
21 views

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

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1answer
39 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
14 views

Calculus scenario involving instantaneous and speed (sequences) [on hold]

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
25 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 ...
2
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1answer
28 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
51 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
13 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
18 views

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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0answers
25 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
22 views

two set of ordinary differential equations

Can you please check my calculation below. Thanks
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1answer
90 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
24 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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0answers
26 views

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} + ...
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
28 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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7 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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0answers
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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
913 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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0answers
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
32 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
15 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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0answers
20 views

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 ...
2
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1answer
28 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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15 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 ...
3
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0answers
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 ...
3
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0answers
21 views

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, ...