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

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1answer
21 views

How many rabbits left

The number of rabbits in a farm increases at a rate proportional to the number of rabbits at a certain time. The number of rabbits doubled to 10000 from the beginning of the year 1985 until the ...
0
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2answers
16 views

Solving second order linear differential equations with non-constant coefficients

Can any one tell me what is the general method to solve the second order differential equation like this: $$ t(t + 1) y '' + (2 - t^2) y ' - (2 + t) y = (t +1 )^2$$ If the general method is ...
-1
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0answers
14 views

Adomian method (how was the solution in this problem obtained?) [on hold]

can someone please help explain to me how the y terms in problem 1 of this paper were obtained in detail paper: http://www.ccsenet.org/journal/index.php/jmr/article/view/45923/24853 thanks here is ...
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0answers
13 views

differential equation spring equation and solve [on hold]

A mass is suspended from a spring. At rest, the mass 3kg stretches the spring .2 meters from its natural length. The mass is then pulled down .4 meters below the equilibrium position and released. ...
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0answers
13 views

find particular solution $y''+y'+4y=t^2+(2t+3)(1+cos(t))$

find particular solution $y''+y'+4y=t^2+(2t+3)(1+Cos(t))$ I'm looking for a way to make the right side less complicated or if there is any trick. What I would do is break it into two parts and add ...
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0answers
21 views

Compute the solutions of the following equation in Fourier space:

$$\frac{d^3u}{dx^3} − αxu = 0, x ∈ R, $$ where $ α > 0$ is some constant and $u(x)$ is assumed to satisfy $\int_R u(x) dx = π.$ I know this is a ODE so this is what I came up with so far: ...
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2answers
46 views

Need a textbook for math course

The undergrad course is called intro the applied math, and it covers: "The unit introduces some of the principal mathematical techniques such as difference equations, differential equations and ...
0
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0answers
11 views

Heat Equation: Remains finite?

I am doing a PDE question. It's about heat equation, spherical coordinates (the usual stuff). The boundary condition is $\frac {\partial T}{\partial r} (1,t) = 0 $ and it also said for $T$ to remain ...
0
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1answer
18 views

Concept of inserting ansatz - separation of variables

In my textbook it says write the unknown function of two variables as a product of two functions of a single variable u(x, t) = X(x) T (t) but then the second step it goes straight away to have T ...
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0answers
16 views

Homogeneous Dilation of the Domain in the Free Membrane Problem

Consider the Neumann boundary value problem of the Laplace operator: $$ \begin{cases} \Delta u+\mu u=0,&\text{in }D,\\ \frac{\partial u}{\partial n}=0,&\text{on }\partial D. \end{cases} $$ Let ...
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0answers
10 views

Solving Partial Differntial Equation with time-dependent Boundary Condition

How does one go about solving differential equations with a time-dependent boundary condition? For example something easy like: \begin{align*} \frac{dh}{dt} &=\frac{d^2h}{dx^2} \\ h(0,t) ...
2
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0answers
34 views

General solution of $ty'+2y=4t^2$

Should we left the general solution of the differential equation $t\frac{dy}{dx}+2y=4t^2$ as $t^2y=t^4+c$ instead of $y=t^2+c/(t^2)$ ($c$ is an arbitrary constant)? Does the solution $y=t^2+c/(t^2)$ ...
1
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1answer
61 views

Numerical series

Consider the series below that consist of 2 different formula $P_aV_a^{1.4}=P_bV_b^{1.4} $ and $P_aV_a=P_bV_b$ that keep repeating itself in the whole sequence. By assuming $P_1$ and $V_1$ both=1, ...
2
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0answers
21 views

Did I correctly derive the scheme for this PDE using the Crank Nicolson Method?

I'm taking an Applied Numerical Methods course this semester, and I was given the following homework problem: Basically, before I begin writing any sort of code, I would like to ensure that I have ...
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0answers
15 views

Convolution - Laplace transform [on hold]

A friend of mine asked me for help. He needs these three exercizes about convolution: $$ \int\,\dfrac{ds}{s(s-1)} $$ $$ \int\,e^{3t} * \sin\,(5t)\,dt $$ $$ \int\,8t^2 * e^{8t}\,dt $$ When I ...
2
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2answers
32 views

How to prove linearity?

Let suppose third-order differential equation, that solved for highest derrivative admits solution: $Y(t) = y(t) + C_1 f_1(t) + C_2 f_2(t) + C_3 f_3(t),$ where $y(t)$ is some solution, $f_1(t), ...
1
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1answer
18 views

differential equation power series solution

I am trying to solve this equation using power series $$ (1-x)y"-xy'+y=0 $$ Knowing that $y(0)=-2$ and $y'(0)=6$. Please I need someone's help, I get a relation between $c(n)$,$c(n+1)$, and ...
0
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1answer
23 views

Equilibrium and Stability of Nonlinear Interactions

Examine the nonlinear model: $$\triangle x_t = rx_t(1-\frac{x_t}{K})-sx_ty_t$$ $$\triangle y_t = -dy_t+\epsilon x_ty_t$$ Find the equilibrium and their stability. Here all the parameters are ...
0
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2answers
22 views

Clarification about correct steps to follow for second order ODE

I have to solve this Cauchy's Problem: $$\begin{cases}y''-y'+3y=x^2-x+3\\y(0)=y'(0)=0 \end{cases}$$ But I have a doubt about the correct steps to follow. It was told me that a second-order ODE is ...
1
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1answer
24 views

Show that all solutions remain in the interval for all time

I really have no idea on how to get started with these, there's no similar example in my book. Do I need to compute $\frac{dy}{dx}$? Any help would be greatly appreciated. Maybe there's just some ...
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0answers
14 views

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 ...
2
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0answers
33 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
9 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
27 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), ...
0
votes
1answer
19 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
40 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,. ...
3
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3answers
52 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 $a_1 = 0$ ...
2
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0answers
16 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
10 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
22 views

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

n-sphotos-h-a.akamaihd.net/hphotos-ak-xta1/v/t34.0-12/11116109_10206718706905332_835173146_n.jpg?oh=baf1ad15e0f70703e5eb93818b61c9d1&oe=55401033&gda=1430237746_5bfaae4f8271730ad293b579ab0e93ab ...
0
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1answer
46 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
16 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 ...
2
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3answers
37 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
37 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 ...
1
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2answers
53 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} ...
0
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1answer
14 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
46 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? $$2y^{(5)}-7y^{(4)}+12y'''+8y''=0$$
1
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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 ...
2
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0answers
28 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
0
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1answer
102 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 ...
1
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1answer
20 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 ...
0
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2answers
27 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 ...
0
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0answers
56 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
votes
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
13 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 ...
1
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1answer
36 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 ...
0
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1answer
11 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?
0
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1answer
14 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? ...
0
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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?