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

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General solution of boundary value problem

I have to find the general solution of the following boundary value problem with the use of Fourier method. $$u_t(x,t)-u_{xxt}(x,t)-u_{xx}(x,t)=0, 0<x< \pi, t>0\\u(0,t=0),t>0$$ ...
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8 views

Fuchs type equation

How to show for any second order equation $u''+p(z)u'+q(z)=0$, with finitely many singularities at $z_0,\ldots,z_n,\infty$ of Fuchs type is of the form $$p(z)=\sum_{j=0}^n\frac{p_j}{z-z_j}, \quad ...
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0answers
10 views

Upper and lower bounds for functional series

Suppose $x\in[0,a]$, $a>1$. Let $g_0(x)\equiv x$, $g_1(x)=(1+x)/2$, and $g_{n+1}(x)=g_1(g_n(x))=g_n(g_1(x))$. Consider $\{\zeta_i(x)\}_{i\ge0}$ where $\zeta_i(x)$ is defined on the interval ...
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0answers
9 views

Prove that there is at most one solution with Green's identity

Prove with the use of Green's identity that the boundary value problem $$\frac{\partial}{\partial{x}} \left( (1+x^2) ...
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1answer
19 views

Why do these equalities stand ?

In my notes there is the following theorem: Let $X_k : [a,b] \rightarrow \mathbb{R}$, $k=1, \dots , n$ an orthogonal system of functions and $X: [a,b] \rightarrow \mathbb{R}$, then $\forall c_1, ...
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0answers
22 views

How do I write this equation as a tridiagonal matrix to write the $n+1$ implicit formula?

I am doing a homework problem for my Applied Numerical Methods class, and I've worked the problem up to this point: $$ \large \frac{u_m^{n+1} - u_m^n}{k}=\frac{u_{m+1}^{n+1} - 2u_{m}^{n+1} + ...
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1answer
20 views

How would one tell if the following ODE has any singular solutions?

The ODE is $y'(x) + \frac{y(x)}{x}=-x^4y(x)^3$. I found the solution to be: $$y(x) = \frac{1}{x\sqrt{\frac{2}{3}x^3+C}}$$ but I'm not sure what is meant by "singular" solutions. Thanks!
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2answers
21 views

Impulse function and the laplace transform

How do I get to the following inverse Laplace transform? $$\mathcal{L}^{-1}\left\{e^{-5s} \cdot \frac 1{s+1}\right\}=u_5(t)e^{-(t-5)} \; ?$$ Here $u_5$ is a step function. I'm using my Laplace ...
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0answers
21 views

Why is the problem in polar coordinates in that form ?

We have the initial and boundary value problem $$u_{xx}(x,y)+u_{yy}(x,y)=0 , x^2+y^2<1 \\ u(x,y)=0 \\ u(1, \theta)=\sin{\theta}, 0< \theta< \pi$$ $$U_{\rho \rho}(\rho, \theta)+ ...
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2answers
16 views

How do I evaluate the Wronskian for this equation

Martin Braun - Differential equations and their applications Chapter 2.1 p.137 Let $y_1,y_2$ be solutions of Bessel's equation $t^2y'' + ty' + (t^2-n^2)y=0$ on the interval $(0,\infty)$ with ...
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1answer
19 views

How have we found the conditions of the problem from the graph?

In my notes there is the following : $$u_{xx}(x,y)+u_{yy}(x,y)=0$$ $$u(x,0)=f(x), 0 \leq x \leq l \\ u(0,y)=0, u(x,\pi)=0 \\ u(l,y)=0$$ How have we found these conditions from the graph?? ...
3
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2answers
46 views

Why are the eigenfunctions linear independent?

At a Sturm-Liouville problem how do we know that the two eigenfunctions that we have found are linear independent?? For example we have the following problem : $$X''+\lambda X=0 \\ X(0)=X(2\pi) \\ ...
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2answers
31 views

Differential equation problam

I have been stuck trying to solve this particular ODE IVP which is: $$ \frac{dy}{dt} = 1+(t-y)^2, \qquad t\in [2,3] \text{ and }y(2)=1 $$ Thanks in advance :)
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2answers
28 views

Let $\lim_{t\to\infty}b(t) = 0$. Show: All solutions of $\dot{x}(t) + x(t) = b(t)$ converge to $0$.

Assignment: Let $b: \mathbb{R} \rightarrow \mathbb{R}$ be continuous with $\lim_{t\to\infty}b(t) = 0$. Show that all solutions for the ODE $$\dot{x}(t) + x(t) = b(t)$$ converge for $t\to\infty$ ...
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0answers
7 views

Periodic solution differential equation

I want to prove, that the solutions $\theta(\tau) = c_1 \cos\tau + c_2 \sin\tau, \quad \tau \in \Re$ to the differential equation $\frac{d^2\theta}{d\tau^2} = -\theta$ are periodic. What would be the ...
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2answers
13 views

PDE Solving: Difference between Similarity Solution and Characteristics?

As far as I understand, both the method of characteristics and similarity solutions allow us to reduce certain partial differential equations to ordinary differential equations which can then be ...
1
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1answer
26 views

Wronskian has constant sign

I don't follow the part in green. Surely if $W(x)$ is zero at some point at say some $c \in (a,b)$ this just implies that $W'(c)=0$ as $p>0$?
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0answers
28 views

ODE: $x' = x+x^2+x^3t$, $x(2)=x_0$. Find $\frac{\partial x}{\partial x_0}|_{x_0=0}$

Problem: $x'=x+x^2+x^3t$ and $x(2)=x_0$. Find $\frac{\partial x}{\partial x_0}|_{x_0=0}$ My attempt: Multiplying both sides by $x^{-3}$ and substituting $y=x^{-2}$ gives us Riccati equation: ...
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1answer
9 views

roots of a linear DE $(D^2+2cD+k)y=0$ given $c<0,k>0, c^2>k$

Let y(x) be a non-trivial solution of the second order linear differential equation $$\dfrac{d^2y}{dx^2}+2c\dfrac{dy}{dx}+ky=0 $$, where $c<0,k>0, c^2>k$. Then, (a) $|y(x)|\to\infty$ as ...
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1answer
12 views

The differential equation of the system of circles touching the y-axis at the origin is

The differential equation of the system of circles touching the y-axis at the origin is (a) $x^2+y^2-2xy\dfrac{dy}{dx}=0$ (b) $x^2+y^2+2xy\dfrac{dy}{dx}=0$ (c) $x^2-y^2-2xy\dfrac{dy}{dx}=0$ (d) ...
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1answer
31 views

How to turn a system of first order into a second order

So I have two equations $X' = aX + bY$ $Y' = cX + dY$ I want to convert it back to a second order equation with the form $X'' + \alpha X' + \beta X$ with $\alpha,\beta$ in terms of a,b,c,d. I ...
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0answers
12 views

Verification of argument for global asymptotic stability of equilibrium point

I am working on proving that an equilibrium point of a two-dimensional dynamical system is globally asymptotically stable. The background and justifications are below. I have gotten to the final steps ...
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0answers
14 views

Solving a system of equations using operator method/notation

The system of differential equations is $$ \left( \begin{array}{ccc} x \\ y \\ \end{array} \right)' = \left( \begin{array}{ccc} 1 & -3 \\ 3 & 7 \\ \end{array} \right) \left( ...
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1answer
16 views

Given a general solution $y_1,y_2$ of $y''+p(x)y' + q(x)y = 0$ take $y_2=v y_1$ and show $\frac{dw}{dx} + (2 \frac{y_1'}{y_1}+p)w=0$ for $v'=w$

Given that $\{y_1(x),y_2(x)\}$ is a fundamental solution set of the ODE $y''+p(x)y' + q(x)y = 0$, I need to show the following: Let the function $v(x)$ be such that $y_2(x)=v(x)y_1(x)$. Show that ...
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4answers
167 views

What is the tip for this exact differential equation?

$$ xdx + ydy = \frac{xdy - ydx}{x^2 + y^2} $$ I have multiplied the left part $x^2+y^2$ for $x dx + y dy$ getting $$(x^3+xy^2+y)dx+(x^2y+y^3-x)dy=0$$ And the derivative test give me: ...
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1answer
54 views

Foundation calculus doubt

So I have an ODE in the following form: $\frac{dx}{d\text{t}} = f(\text{m}) sin\text{z}$ where z = z(t) and m = m(t) i.e. they are both functions of time, t. Now, if I were to concern It is possible ...
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2answers
24 views

Solving for the constant with initial condition of differential equations.

When solving for the constant in a differential equation, is it fine to check for it at any step without showing the constant manipulations? For example: $$\frac{dP}{dt}=2P(t)(1-\frac{P(t)}{250})$$ A ...
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1answer
31 views

System of ODEs obtained by using the method of characteristics for $u_x + 2u_t - 4u = e^{x+t}$

I have a question which requires me to use the method of characteristics in order to solve the PDE $u_x + 2u_t - 4u = e^{x+t}$. This results in the system of ODE's $\frac{dx}{dr} = 1 , \frac{dt}{dr} ...
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1answer
24 views

Show that the Wronskian of solutions of $y''+p(x)y'+q(x)y=0$ satisfies $\frac{dW}{dx}+pW=0$

So I am given: $\{y_1(x),y_2(x)\}$ is a fundamental solution set of the ODE: $$y''+p(x)y'+q(x)y=0$$ I need to show that the Wronskian $W(y_1,y_2)$ satisfies the ODE $\frac{dW}{dx}+pW=0$ and hence, ...
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2answers
32 views

solution for a first order ordinary differential equation with displacement

I have to described the solutions for the following IVP $$y'(t)=y(kt), y(0)=1$$ where $k$ is a positive constant. I tried to solved it but didn't get anywhere, can anybody give me a clue of how to ...
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0answers
17 views

Continuous time ODE to Discrete time ODE?

So I've been working a bit with a continuous time ODE (the logistic equation) given by: $$\frac{dx_i}{dt} = x_i(b_i-\sum_{j=1}^{n}a_{ij}x_j)$$ I was looking at another paper which looks at the ...
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1answer
19 views

Turn 2nd order ODE in to Sturm-Liouville Form

How Do I turn $$(x+1)y''-xy'+y=0$$ into Sturm-Liouville Form?
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1answer
27 views

Need help with a Crank Nicholson Method example problem.

I have an exam coming up and the professor released the sample test containing a Crank Nicolson question. I was out of town for those two lectures, so I missed the information. Even though I have ...
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1answer
32 views

How many conditions do we need for a problem to have an unique solution?

How do we know how many initial and boundary conditions we need for a problem to have an unique solution ?? For example if we have the problem $$u_{tt}-u_{xxtt}(x,t)-u_{xx}(x,t)=f(x,t), 0<x<1, ...
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2answers
46 views

The reasoning behind variation of parameters.

Let's say you have the second order equation: $y''+p(x)y'+q(x)y=f(x)$ And let's say you have found two solutions ($y_1$ and $y_2$) to the homogeneous equation: $y''+p(x)y'+q(x)y=0$. Then the ...
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0answers
28 views

Initial&Boundary Value problem-Fourier

$$u_{t}=u_{xx}, \hspace{5mm} x>0, t>0$$ $$u(0,t)=0 \hspace{3mm} u(x,0)=f(x)$$ We want that the solutions are bounded. We are looking for solutions of the form $$u(x,t)=X(x) \cdot T(t)$$ ...
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1answer
13 views

Differential system, a matrix with eigenvalue

Let's say that we have $n$ differential equations written in the form: $x'(t) = Ax(t) + v \exp(\lambda t)$, where $v$ is the eigenvector of $A$ such that $A v = \lambda v$ and $A$ is a $n \times n$ ...
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1answer
24 views

Euler method uniform convergence

I have a question for you guys. Given a differential equation $$\dot{x}=f(x)\qquad x\in\mathbb{R}^n$$ on a compact interval $[0,T]$. If one considers for every $k\in\mathbb{N}$, the Euler's ...
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0answers
27 views

Meaning of the Maximal Interval of existence

What does it mean to say that a solution to a differential equation has a maximal interval of existence?
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3answers
56 views

ODE) $y'' + 2y' = 1 + t^2 + e^{-2t}$

I'm stuck with the ODE problem: $y'' + 2y' = 1 + t^2 + e^{-2t}$ This problem is in "judicious guessing" chapter of Braun's "Differential Equations and Their Applications". The trick he taught in ...
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0answers
21 views

Why is solving this differential equation by separation of variables and using undetermined coefficients not the same?

Given $\frac{dx}{dt}=r(x_{0}-x)$, I was able to solve the equation by separation of variables but it is not coming out the same if I were to use undetermined coefficients. Why is that?
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1answer
50 views

Show that $y_1(x) = \int_c^x f(x-t) R(t) dt$ is a particular solution of $L(y) = R$

This is problem 14 from Chapter 6.15 of Apostol Calculus, Volume 2 (p. 167): If $L(y) = y'' + ay' + by$, where $a$ and $b$ are constants, let $f$ be that particular solution of $L(y) = 0$ ...
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1answer
10 views

Question regarding $\mathcal{L}\{t*\mathcal{U}(t-2)\}$

I'm working on a problem for homework (* is multiplication not convolution): $\mathcal{L}\{t*\mathcal{U}(t-2)\}$ I understand that $\mathcal{L}\{(t-a)\mathcal{U}(t-a)\}=e^{-as}F(s)$ The first step ...
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0answers
21 views

Solving laplace equation and decomposing

I want to solve $$ 9U_{xx}+U_{yy}=\sin (2\pi x) + \sin(2\pi y) \label{eq:1}\tag{1} $$ with $U=0$ on the boundary of the unit square. I know you would have to decompose the problem to satisfy each of ...
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3answers
47 views

$\frac{d^2y}{dt^2}+4y=t\sin(2t)$'s particular solution

Finding the particular solution of : $\frac{d^2y}{dt^2}+4y=t\sin(2t)$ Hey everyone! My professor recently went over this problem and I can't seem to find where he derived a particular equation. ...
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0answers
21 views

Why do we take the odd extension?

When we have the initial and boundary value problem $$u_{tt}(x,t)-c^2u_{xx}(x,t)=0, x>0, t>0 \\ u(0,t)=0 \\ u(x,0)=f(x), x \geq 0 \\ u_t(x,0)=g(x), x \geq 0$$ can we apply Green's theorem or ...
4
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2answers
43 views

Question about this ODE? $\frac{dy}{dx} = \frac{2x-y}{x+2y}$

Am I being dumb, or is this question actually hard? I made the substitution $u=y/x \implies y = ux$, so then I get: $x \cdot \dfrac{du}{dx} + u = \dfrac{2x-ux}{x+2ux} \implies x \cdot \dfrac{du}{dx} ...
0
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0answers
21 views

How to solve C'=-a*N'?

I am a Molecular Biologist and I had been watching this video on Bacterial growth. At 9:25, the teacher solves differential equation by integrating: $$ C'= -\alpha*N' $$ Integrating both sides of the ...
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0answers
43 views

Find all differentiable functions $f$ such that $f(f(x))=f'(x)$ [duplicate]

Here is a problem I made up: Find all differentiable functions $f$ from the reals to the reals such that $f(f(x))=f'(x)$ for all real $x$.
0
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1answer
22 views

Having trouble with discretization and boundry value problems

I have the following homework question: Consider the boundary value problem $y''(x) + 5y'(x) − (2 + x)y(x) = e^x$ on $x ∈ (0, 2)$ with boundary conditions $3y(0) + y'(0) = 5$ and $y'(2) = 7$. ...