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

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Stick breaking point (discretized ODE)

I cannot find nontrivial solutions to the following problem. Let $x\in[0,1]$ and $y(x)$ be the deflection of the stick. Then this is described by the diff.eq.: $$\alpha^{-1} P y(x)+y(x)''=0 $$ where ...
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0answers
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Does $-\Delta u\equiv u^p$ have non-positive radial solutions?

Let $p>1$ and $u:[0,R)\to\mathbb{R}$ be a radial solution of $$\left\{\begin{matrix}\displaystyle-u''-\frac{n-1}ru'&\equiv&u^p&&\text{on }(0,R)\\ u'&\equiv ...
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1answer
15 views

Decide the smooth function $r : \mathbb R \rightarrow \mathbb R$ of the equation $r(t)^2 + r'(t)^2 = 1$.

Suppose $r:\mathbb R \rightarrow \mathbb R$ is a smooth function and suppose $r(t)^2 + r'(t)^2 = 1$. I want to determine the function $r(t)$. I see that $r(t)^2 + r'(t)^2 = 1$, so I could take $r(t) ...
3
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0answers
29 views

Solutions for large $x$

Given the differential equation $$y''(x)-x*y(x)+y^3(x)=0,$$ look voor the two following solutions with $x$ large and positive. Look for: a) A oscilatory solution with two arbitrary coefficients b) ...
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1answer
39 views

Is there a unique solution of $\gamma(t)= f''(t)f(t) $ with $f(0) =0$ and $f'(0)=1$?

Consider, \begin{align*} \gamma(t) &= f''(t)f(t) \\ f(0) &=0 \quad f'(0)=1 \end{align*} where $f(t)$ is an unknown function and $\gamma(t)$ is a known function. Is there a unique solution ...
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38 views

This is one from my Diff'Eq Honors (they may see this) [on hold]

One day it started snowing at a heavy and steady rate. A snowplow started out at noon, going 2 miles the first hour and 1 mile the second hour. What time did it start snowing? (Hint: the speed of the ...
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0answers
11 views

diffusion equation [on hold]

I'm kinda lost with this problem. I don't know how to solve it. If somebody can help me I will be so thankfully. I'm so confuse.If somebody know a reference problem that would help a lot
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22 views

heat equation, total heat energy [duplicate]

I'm having a hard time with this problem. I get the situation, but I just don't know how to model it and show part b and part c. I will be so thankfully.
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1answer
22 views

Boundary conditions

I am kinda confuse with the second part of my homework. I did the first part (3/a and 4/a) without any problem, but part b for both problems I don't get it at all. I try to plug the boundaries in the ...
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0answers
22 views

How to solve the vector differential equation? [on hold]

I'm new to this section, so I'm trying to solve vector differential equations, and I need some guidance. Could anybody give a step-by-step process for doing so, so that I could do some more problems ...
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3answers
24 views

Intro to Differential Equations Problem

Show that $y(t)= C_1 e^{2t} + C_1 e^{-2t}$ is a solution to the differential equation $y'' - 4y = 0$. $C_1$ and $C_2$ are arbitrary constants. This was the first part of the problem which I ...
2
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1answer
30 views

Solving $r^2 u_{rr} + 2ru_{r} + r^{2}u = 0$ directly

The problem I am working on boils to solve the differential equation $$r^{2}u_{rr} + 2ru_{r} + r^{2}u = 0.$$ The solution to this equation is the spherical Bessel function $u(r) = \sin(r)/r$. However, ...
1
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2answers
16 views

Elliptic differential operator

I am given the differential operator $D(f):=-(fg)'+hf$ and $D^* (f) = g \cdot f' + hf$ where $h,g$ are some smooth functions and want to find out under which conditions, these two operators are ...
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1answer
17 views

Methods of Solving Ordinary Differential Equations - A Small Question

I've spent some weeks now trying to learn how to solve ordinary differential equations, and I am now studying the Laplace transform and how this can be applied to solve ODEs. I feel a little bit ...
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0answers
15 views

Solving a DE with no initial conditions

I'm having some sort of difficulty on my signals homework. I am given the following problem. Where u(t) is a unit step function. For whatever reason, most of the problems assigned have no initial ...
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1answer
29 views

Showing flows converge in the phase plane

I have a system of ODEs: $$\dot{x} = \frac{m_1 x (1-x-y)}{a_1 + 1-x-y} - x$$ $$\dot{y} = \frac{m_2 y (1-x-y)}{a_2 + 1-x-y} - y$$ I'm trying to show that all the flows converge to the point ...
3
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2answers
42 views

How to solve the following linear differential equation?

I'm having trouble solving the following differential equation: $y'(x)=\frac{8A^2x}{(1+4A^2x^2)^2}\cdot y-4Bx$ $A$ and $B$ are real constants. I would be very grateful for any help. Thanks in ...
0
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1answer
19 views

Show $y_1(t) = y(t)\int^t_{t_0} \frac 1 {x_1(s)^2} e^{-\int_{t_0}^s p(r) dr} ds$ solves the 2-nd order ODE: $x'' + p(t)x' + q(t)x = 0$

Suppose $(I,y)$ solves the 2-nd order ODE: $x'' + p(t)x' + q(t)x = 0$. Assume $y(t) \neq 0$ for $t \in I$ and let $t_o \in I$. I want to show that $(I, y_1)$ where $$y_1(t) = y(t)\int^t_{t_0} \frac ...
2
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1answer
19 views

Eigenfunctions of the laplacian (1 dimension)

I have the following problem: $\frac{d^2 u}{dx^2}(x)+\lambda u(x)=0, x \in (a,b)$ and $u(a)=u(b)=0$. The general solution (for $\lambda>0$) is $u(x)=c_1\cos(\sqrt\lambda x)+c_2 \sin (\sqrt\lambda ...
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2answers
18 views

Finding the value of a constant given an equation where the sum of the roots is -3

I am to find the value of h given the equation 3hx^2 - 2x +5xh = 3. The sum of the roots of the polynomial is -3. I am having ...
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0answers
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How to find first-order quasi-linear PDEs form second-order quasi-linear PDE?

Transform $u_{tt} u_{xx}-u^{2}_{tx} + uu_{tt} + 1=0 $ into first-order quasi-linear PDEs. Attempt: $u_{tt}(u_{xx}+u)=(u_{tx}-1)(u_{tx}+1)$ To get $u_{tt} = u_{tx}-1\Rightarrow u_t = u_x ...
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0answers
20 views

Verify that $e^{at}$ is the only solution to the ODE: $y' = a y$ defined on $\mathbb R, a \neq 0$

Find all solutions to the ODE: $y' = a y$ defined on $\mathbb R, a \neq 0$ By inspection, I see that $e^{at}$ is a valid solution. However, my problem is to verify that $e^{at}$ is the only ...
0
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1answer
57 views

Why is -ln x is not equal to 1/ln x?

I am doing differential equation now and I need to convert them into the proper form in order to do my homogeneous differential equation. So now I just found out that -ln x is not equal to 1 / ln x. I ...
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0answers
28 views

solve $y(x)+\int_{0}^{x}(x-s)y(s)ds=\frac{x^{3}}{6}$ [on hold]

Let $y:[0,\infty) \rightarrow \mathbb{R}$ be twice continuously differentiable and satisfy $$y(x)+\int_{0}^{x}(x-s)y(s)ds=\frac{x^{3}}{6}$$ then which of the following is true 1. ...
0
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1answer
79 views

Solving a first order linear ODE and determining the behavior of its solutions

(a) Draw a direction field for the given differential equation. How do solutions appear to behave as $t → 0$? Does the behavior depend on the choice of the initial value $a$? Let $a_{0}$ be the value ...
3
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1answer
43 views

Fredholm Integral Equations - Sturm-Lioville & Green Function Theory?

In an ODE's book one is given a 2nd order ode boundary value problem like $$y'' + A(x)y' + B(x)y = f(x), y(a) = y_a, y(b) = y_b$$ and might be told to analyze it with a Green function or via ...
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0answers
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what is degree of given PDE…

When the given differential eqn is completely free from radicals the the final exponent on the highest order derivative amounts degree of given differential eqn. In present case it is 3 or 6. i.e. ...
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1answer
48 views

Set up differential equation

As people get older, they perceive time differently. The older one is, the faster time goes by. To quantify this issue, we create a model: The entire perceived period of time shall be $w(t)$ . A ...
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2answers
28 views

Mathematical Puzzle: A Drag Race of Who Wins

I'm having a real difficult time understanding how this problem is solved: "Two drivers, Alison and Kevin, are participating in a drag race. Beginning from a standing start, they each proceed with a ...
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Properties of the solution of an initial value problem

I have an IVP which can not be solved for explicitly of the form: $y''(t) = f(y)(y')^2 +ay'-g(y)$ $y(0)=0, y'(x_1) = h(y)>0$ with $y,y',y'' \in \mathbb{R}_+$, $x \in [0,x_1]$ and I know that the ...
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1answer
42 views

Given a solution of a differential equation, determine the differential eqution itself

Sorry if my layout is bad, I'm new. So this question was asked a couple of years ago on an exam about differential-equations. Suppose you have a third order differential-equation with the following ...
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2answers
35 views

How to solve differential equation $\frac{d}{dx}\left(\frac{\lambda y'}{\sqrt{1+y'^2}}\right)=1$

My task is to solve for $y$ from: $$\frac{d}{dx}\left(\frac{\lambda y'}{\sqrt{1+y'^2}}\right)=1$$ I have been given the answer, but I would like to calculate this myself also. $\lambda$ is a ...
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1answer
36 views

differential equation degree doubt

$dy/dx = sin^{-1} (y)$ this is a form of $dy/dx = f(y).$ so degree should be $1.$ but if i write it as $y = \sin(dy/dx).$ then degree is not defined as it is not a polynomial in $dy/dx $ please ...
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1answer
61 views

Checking: finding extremals for a functional

I'm trying to find the extremals of the functional $$J[y] = \int_0^1 (y')^2 + y^2 + 4ye^x \, {\rm d}x,$$ imposed that $y(0) = 0$ and $y(1) = 1 $. I got that there can't be extremals, and that's weird ...
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0answers
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monotonicity of a $C^2(\mathbb{R})$ function

Let $c>0$ and $u(\xi)\in C^2(\mathbb{R})$ be a solution of $$ (D(u)u')'+cu'+g(u)=0,\qquad '=\frac{d}{d\xi} $$ with $c$. The assumptions for $D$ and $g$ are respectively $$D\in C([0,1])\cap ...
2
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1answer
24 views

the boundary value problem: $u''(x)+\lambda u(x)=0,x\in (0,1),$ $u(0)=u(1); u'(0)=u'(1).$

Find all possible $(\lambda,u)$ where $\lambda \in \mathbb R$ and $u\ne0$, to the boundary value problem: $u''(x)+\lambda u(x)=0,x\in (0,1),$ $u(0)=u(1); u'(0)=u'(1).$ My Effort: for ...
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0answers
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Differential equation in Maple : No solution on $x = -1 .. 1, y = -1 .. 1$.

Backround: Yesterday in class we had a lab session (practical work ?) on ODE and I have a question. We plot the following contour (I am using maple) implicitplot(H(x, y) = 0, x = -1 .. 1, y = -1 .. ...
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1answer
21 views

Cannot figure out a second order lineary differential equation with initial values

I got the following question: Solve the following initial value problem: $y(0) = 0$, $y'(0) = 1$, $$y'' + 10y' + 25y = 0$$ So I started with getting the general solution: $$ y(x) = C_1e^{-5x} + ...
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1answer
65 views

Solving a Linear ODE

Solve the following linear ODE $$3t^{2}y'+t^{3}y=cos(t)$$ What i tried Since this is a linear equation, i used the integrating factor method. First i didide both the LHS and RHS of the equation by ...
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0answers
21 views

Example: Solve a Second Order Nonhomogeneous ODE with Constant Coefficients by Variation of Parameters (2R-17)

Problem to solve: $$(D^2-2D+1)y=\frac{e^x}{x^3}$$ Answer in text: $$y=(c_1+c_2x)e^x+\frac12\frac{e^x}{x}$$ Our solution begins by rewriting the ODE in a more familiar form: ...
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1answer
36 views

Study of systems of Linear Differential Equations?

Is there any area of mathematics that deals with and formalizes systems of Linear DEs, akin to how Linear Algebra deals with systems of linear equations? Does it provide any insightful results?
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1answer
29 views

Linearity of a differential equation

The following is the general form of a linear ODE, where $t$ is the independent variable and $y$ is the dependent one: $a_n(t) \frac{d^ny(t)}{dt^n} + a_{n-1}(t) \frac{d^{n-1}y(t)}{dt^{n-1}} + \dots + ...
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0answers
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Connection between the Laplace transform and generating functions

As I was sitting through a boring lecture rehashing basic techniques to solve ordinary differential equations, I began thinking about the Laplace transform and scribbled down a few ideas that I've ...
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0answers
32 views

Differential Equation Direction field

What i want to achieve: I want to plot the direction fields of the following three differential equations: 1. Malthusian growth model: $p'(t)=\lambda*p(t)$ with $\lambda=1$ and $p(t)=t$ 2. Linear ...
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0answers
35 views

How to show that a leaf is topologically a cone.

I am trying to understand the topological behaviour of foliations around irreducible singularities, specially in the case of singularities in the Poincaré domain. I am using the third chapter of this ...
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1answer
38 views

Eigenvalues and eigenfunctions of fourth order ODE

Find the eigenvalues and eigenfunctions of the problem $$y^{(4)} − λy = 0$$ with the boundary conditions (i) $\quad y(0) = y'' (0) = y(β) = y'' (β) = 0$ (ii) $\quad y(0) = y' (0) = y'' (β) = y''' ...
2
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1answer
37 views

Inverse of $3$ by $3$ matrix with non-constant entries.

I'm solving a question in nonhomogenous ordinary differential equation system $x'=Px+q$, and to solve my question I need to compute the inverse of the matrix $A=\begin{pmatrix}e^{-2t} & e^{-t} ...
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1answer
38 views

Why do we need sturm liouville form to solve ODE?

What is the reason that we have to recast a 2nd order ODE into SL form to find its eigenfunctions? for example, let $Ly=y''+y'+\frac{y}{4}=-ky$, boundary conditions $y=0$ at $x=0$ and $y-2y'=0$ at ...
0
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0answers
32 views

integral equation into differential equation

I have the equation $$ E = \alpha \int \int_S E dS $$ and I need to find a solution for E. My first instinct is to re-arrange it into a second order differential equation, but because dS is an area, ...