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

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2
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6 views

Laplace equation between circles

I need to solve the simple Laplace equation $$\nabla^2f(r,\theta)=0$$ with boundary conditions: $$f(a,\theta)=g(\theta)$$ $$\lim_{A\rightarrow\infty}f(A,\theta)=1$$ where $a$ is a fixed real radius. ...
0
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0answers
17 views

A few queries of the method of variation of parameters

I've been reviewing my knowledge on the technique of variation of parameters to solve differential equations and have a couple of queries that I'd like to clear up (particularly for 2nd order ...
1
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0answers
16 views

Solving nonlinear differential equation using boundary value at infinity

I want to solve the following differential equation subject to the condition that $f(0)=0$ and $\lim_{x\rightarrow\infty}f(x)=1$. Also $|f| < 1 $ always. Can anybody suggest me a concrete way ...
0
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1answer
12 views

getting a new differential equation from an old one.

Suppose I have the following logistic differential equation: $$f'(x) = f(x)(1-f(x)), f(0) = 1/2 $$ and suppose that $ x = 2y - a$ for some positve constant $a$. How do I write a differential ...
0
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0answers
8 views

Comparison theorem for parabolic partial differential equations

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain $J\subseteq\mathbb{R}$ be an intervall $T\in(0,\infty)$ and $f\in C^0\left(\overline{\Omega}\times[0,T]\times J\right)$ be locally Lipschitz ...
0
votes
2answers
22 views

general solution of second order linear de

Let 1, x and $x^2$ be solutions of second order linear non homogeneous differential equation $-1\lt x\lt 1$. Then find the general solution. I only know that general solution is sum of complementry ...
0
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1answer
27 views

First-order nonlinear differential equation

How would I solve this differential equation for $y(x)$? $\frac{dy}{dx} = \frac{y-xy}{x-xy}$ $y -\ln(y) = x - \ln(x) + C$ I'm not sure what to do at this point. I looked it up on WolframAlpha and ...
3
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2answers
134 views

Limit of solution of differential equation without solving the equation.

Given $$x'(t)=A-B\left(x(t)\right)^2, \quad x(0)=0.$$ Is it possible to find $\lim\limits_{t\to\infty}x(t)$ without solving the differential equation? Assuming $\lim\limits_{t\to\infty}x'(t)=0$ gives ...
0
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1answer
29 views

General solution to diffeerential equation

Given the differential equation $$\frac{dy}{dt}=\frac{4t}{1+3y^2}$$ is this the general solution? $$y+y^3=2t^2+c$$ Can we continue to simplify it?
2
votes
1answer
22 views

Extracting differential equations [duplicate]

$$\frac{dx}{dy} = \frac{x(\alpha - \beta y)}{y(\delta x - \gamma)}$$ How do I extract two differential equations (y as a function of x and x as a function of y) from the equation above? I could ...
1
vote
2answers
26 views

Can a differential equation with real coefficients have solution with complex coefficients?

Can a differential equation (with constant coefficients, linear or nonlinear) with real coefficients have solution(s) with complex coefficients? If so, are there any examples related to actual ...
1
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1answer
38 views

Reduce this third order ordinary differential equation to first order to use Runge Kutta

The ODE I'm working with is $$\dddot{x} + t^2\ddot{x} + 4x = 0$$ with $$x(0)=1, \dot{x}(0)=0, \ddot{x}=-1$$ I've written a very basic program in C++ to use the RK4 method to approximate a solution to ...
0
votes
3answers
167 views

Eigenvalue of some Sturm–Liouville problem

I have one simple question. How I suppose to show that $\lambda =0$ is an eigenvalue of some problem. Does it mean that I must have non-trivial solution for $\lambda=0 $? Thanks! UPD:I mean by ...
0
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1answer
39 views

Showing a system is fully self adjoint for general unmixed boundary conditions

I have been asked to look at the following questions and I'm struggling to solve it. Let $Ly=a_2(x)y''(x)+a_1(x)y'(x)+a_0(x)y(x) , a<x<b$ such that $L^*=L$. i.e. $L$ is a self adjoint linear ...
0
votes
1answer
22 views

Show that this equation together with the boundary conditions $u(0) = 2, u(\pi) = 0$ has no solution

Consider the ordinary differential equation: $u'' + u = 0$. I have no idea how to solve this, no idea what so ever. Please help.
0
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1answer
396 views

Cauchy Peano Existence Theorem

http://www.math.iitb.ac.in/~mcj/root.pdf I was going through the above link and in the section 3.4 to prove the Cauchy Peano Existence Theorem they first prove the set of functions is relatively ...
0
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1answer
19 views

Verifying transport equation solution

I have just started PDE's and I have the transport equation $u_t + au_x = 0$ which has the general solution $u(x,t) = f(x - at)$ In a book I'm reading it says this can be verified by substitution ...
0
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0answers
8 views

Second order linear ODE arising from Euclidean heat kernel

When solving for the Euclidean heat kernel $H(t,x,y) \in C^{\infty}((0,\infty) \times \mathbb{R}^n \times \mathbb{R}^n)$, one way to proceed is to look for a solution in the form $H(t,x,y) = ...
0
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0answers
20 views

Is it possible to show the uniqueness of formula for solution?

The motivation to this question can be found in: Show that any sequence $(u_{n})$ must tends to infinity as $n→∞$ My question is: Is it possible to show the uniqueness of the formula for the ...
0
votes
1answer
45 views

How to solve: y'' + 9y = sin(3t)

I need to find the particular solution to the equation: $$y'' + 9y = \sin(3t)$$ I thought we were looking for a trigonometric forcing term on the form: $$y = a\cdot\cos(3t) + b\cdot\sin(3t)$$ But ...
1
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1answer
19 views

Help in solving linear differential equation.

The equation is: $(xy^4 + y)dx -xdy =0$ I brought the differential terms to the same side and then divided by $y^2$ to get this. $(xy^2)dy=d(y/x)$. I tried an alternate way to simplify it which ...
1
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1answer
24 views

Value(s) of the parameter $a$ that give explicit formula's

For what value(s) of the parameter $a$ is it possible to find explicit formula's (without integrals) for the solutions to $$\frac{dy}{dt}= aty +4e^{-t^2}$$ The answer is $a=-2$. I don't know how to ...
1
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1answer
355 views

Wrong answer for this differential equation temperature problem.

(a) An object is placed in a 68°F room. Write a differential equation for H, the temperature of the object at time t. ANSWER: dH/dt = -k(68 - H) (b) Give the general solution for the differential ...
6
votes
1answer
578 views

Damped Harmonic Oscillator and Response Function

This is another one of those questions that I feel like I am almost there, but not quite, and it's the math that gets me. But here goes: For a driven damped harmonic oscillator, show that the full ...
0
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0answers
29 views

Fokker-Planck equation - find probability density function

I have problem from my course, that I can't solve. If anyone can do it and explain, would be great. Find the probability density function $f(x,t)$, of $X_t$ where {$X_t$} is a solution of stochastic ...
0
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0answers
19 views

Substitution in a system of ordinary differential equations when terms of the same order derivative for different variables occur in the same equation

Let's say I have a differential equation such as: y'' - 2ty' + y = 0, y(0) = 2.1, y'(0) = 1.0 I can solve this (among other ways) by substitution and conversion ...
2
votes
1answer
27 views

Why aren't my Laplace transform and Undetermind Coefficients answers matching up?

I might be losing my mind this morning (I am, for sure), but I can't these two techniques to give me the same answer to a basic differential equations problem. The problem is $y''-8y'+27y=0$ with the ...
1
vote
1answer
23 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) ...
0
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0answers
27 views

Solution of a non-linear ODE system

I'd like to find an explicit solution for the following system of ordinary differential equations: \begin{cases} \frac{dx}{dt}=-x+\frac{ax}{1+x}+\frac{by}{1+y}+c\\ \\ ...
0
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1answer
39 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 ...
0
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0answers
14 views

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 ...
0
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0answers
13 views

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 ...
1
vote
1answer
46 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 ...
1
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2answers
17 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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0answers
12 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
0
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0answers
24 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.
0
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1answer
24 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 ...
3
votes
1answer
46 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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votes
0answers
26 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 ...
3
votes
2answers
46 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 ...
2
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1answer
33 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, ...
0
votes
3answers
29 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 ...
1
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1answer
24 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
16 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 ...
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
votes
1answer
21 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 ...
0
votes
2answers
19 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 ...
2
votes
0answers
34 views

Seeking good parametrization for the homoclinic solution

Could somebody quickly provide me with a good parametrization for the homoclinic solution $$\frac{p^2}{2}-\frac{q^2}{2}+\frac{q^3}{3}=0$$ of the system \begin{aligned} \dot{q}&=p\\ ...
1
vote
0answers
11 views

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 ...