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

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1answer
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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
28 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
82 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
29 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
17 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
38 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
17 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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1answer
19 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 [closed]

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
17 views

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
2k 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
30 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
27 views

Help with a Differential Equations problem. [closed]

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
36 views

Solving a system of coupled differential equations [closed]

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
23 views

Solving ODES in PDE

The PDE given as: $$t^2u_t-\text{yu}_x+\text{xu}_y\text{=0}$$ The characteristic equations are: $$\frac{\text{dt}}{\text{dt}}=t^2$$ $$\frac{\text{dx}}{\text{dt}}\text{=-y}$$ ...
0
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1answer
37 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 ...
0
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1answer
22 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
16 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?
1
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1answer
24 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
34 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
35 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 ...
1
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1answer
33 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
17 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
64 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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0answers
23 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
81 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
33 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
56 views

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

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

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

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 ...
2
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2answers
50 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 ...
0
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1answer
11 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 ...
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2answers
10 views

Time dependence of velocity from position dependece of velocity

I know dependence of velocity on position $v(x)$ and I wan't to know dependence of velocity on time $v(t)$ I was thinking that using some chain rules or derivative of inverse it would be possible to ...
0
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1answer
23 views

Solve the Initial Value Differential Equations

I split the equation and got y+1 dy = xysinxdx, then I divided the right side by y to get 1 + (1/y) = xsinx dx. I took the integrals of both sides and got y + lny = -xcosx + sinx + c. I don't ...
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0answers
23 views

Second order linear ODE and undamped

I am a bit confused with this problem: An object with mass 1 slug is attached to a vertical coil spring of spring constant of 1 pound per foot. After coming to equilibrium, the object is set into ...
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1answer
33 views

quick question with 2nd order linear differential equations

I am solving $y''+4y'+5y=2e^{-2x}cos(x)$ I am working on determining $A$ and $B$ in the particular function. I have the following 2 equations: for the sine part : $-2A+3Ax-3B+Bx=0$ for the ...
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2answers
36 views

Fourier series of complex diff eq

Can I just use Euler's identity to construct the Fourier Series since it is complex? I was personally thinking I could, but I wanted to be doubly sure.
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0answers
14 views

For what types of differential equations is the Laplace transform most effective?

I'm reviewing for a final exam and want to make sure I know what tools to use for what situations, and was just wondering if there were situations where the Laplace transform is unusable or less ...
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1answer
30 views

Need help for this case:

I am learning the artificial potential field method for path planning of mobile robot; artificial potential field method has two components: the first one is attractive force and second one is ...
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1answer
33 views

Topological structure/graph from a paper

This question is based off a paper titled "On designing heteroclinic networks from graphs." I'm having a difficult time visualizing something "drawn in 4-dimensions" projected down to a 2-dimensional ...
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1answer
40 views

Help solving differential equations

I would like to know how to classify the following equations: $y''+ 4y'+5y=2e^{-2x}cos(x)$. Is it a second order linear equation?
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1answer
51 views

Can the following nonlinear first order ODE be solved?

I have tried solving this equation from several manners but no luck. Can it be solved? $$\frac{d f}{d t} = A f^2 +g(t)$$ The solution for the homogeneous is (I think; somebody should confirm) ...
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0answers
39 views

Lotka-Volterra Problem From Arnold's Ordinary Differential Equations

Problem 1 of section 2.7 of Arnold's Ordinary Differential Equations book asks to prove that the period of the oscillations in the Lotka-Volterra model tends to infinity as the initial condition ...
4
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1answer
46 views

Did I do something wrong solving this PDE in MATLAB?

I have the following PDE problem on a practice exam: I have completed the problem using MATLAB to the best of my ability. Here is the code I used ...
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0answers
72 views

Why does $\frac{1}{r}\frac{dr}{d\theta} = \cot \psi$?

In the discussion of linear fractional equations in Birkhoff and Rota's Ordinary Differential Equations, the authors assert that if we convert a DE of the form $y' = F\left(\frac{y}{x}\right)$ to ...
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1answer
35 views

Is there any nonnegative $u\in C^2(\mathbb{R}^n)$ with $-\Delta u=1$ in $\mathbb{R}^n$?

Is there any nonnegative $u\in C^2(\mathbb{R}^n)$ with $\Delta u=-1$ in $\mathbb{R}^n$? I think not, but how can we prove it? Let's assume that such a solution exists. Let $R>0$ and $B_R:=B_R(0)$ ...
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0answers
14 views

Ordinary Differential Equations self-study reference request

I know there are a lot of reference requests for differential equations textbooks but none seem to be what I need. I'm looking for a book I can use for self study that isn't overly complicated and ...
0
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1answer
28 views

solving a partial differential equation

How can I solve the following equation? $$-f_{x}+yf_{xy}+xf_{yy} = c^{'}(x)(-f+yf_{y})$$ where $f=f(x,y)$ is a real function of two variables $x,y$ and $c=c(x)$ is a real function of $x$. I guess ...