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

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Given a system of differential equations, how can one tell if $\textbf{x}_c = (0,0)^T$ is a unique critical point?

I have: $$\frac{d\textbf{x}}{dt}=\begin{bmatrix} -1 & 2 \\ -2 & -1 \\ \end{bmatrix}\textbf{x}(t)$$ with $\textbf{x}(0)=(1,-1)^T$. I am asked whether the critical point ...
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20 views

Proof for Cauchy Problem solution in $\mathbb{R} \to\mathbb{R}^n$

I want to verify if I made this proof correctly (original version in spanish so please excuse my translation mistakes)... Let $f: \mathbb{R} \to\mathbb{R}^n$ continuous and Lipschitz on second ...
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1answer
31 views

Zeros of a differential equation

Is the part highlighted in green correct? Could there not be infinitely many zeros in the region $x<N$?
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8 views

How is an ODE a consistency condition?

I was reading a text on Optimal Control Theory by E. Todorov, when I came accross this passage (on page 10): An ODE is a consistency condition which singles out specific trajectories without ...
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2answers
20 views

Differential equations in function

Equations (1) : $xy'+(1-x)y=1$ let $z=xy+1$ determine and solve the differential equation (2) whose general solution is the function $z$ . -determine the general solution of (1)
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Shifting limits in series solutions to ODEs

I'm trying to practice the Frobenius method of solving ODEs, and I keep getting the answer wrong. It seems to be down to the shifting of limits of the sums, although it is not clear in the solutions I ...
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36 views

ODE with strange initial condition

Problem: Let $f:\mathbb{R}\to\mathbb{R}$ be a differentiable function and $k_0$ be it's fixed point (not necessarily the only one). Prove that: $f'(k_0)<1$ implies that there are no ...
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1answer
41 views

What kind of differential equation is $(x^2+2y^3)y'=xy$?

what kind of differential equation is $(x^2+2y^3)y'=xy$?, I think its an inexact differential first order, its surely not linear, I tried to check if its separable also didn't work, its not also ...
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44 views

$y=\cos(m \arcsin x)$ Validity of solution $\dfrac {dy} {dx}$ when $x=0$?

$y=\cos(m \arcsin x)$, for $ -1 < x < 1$ I want to find the value of $\dfrac {dy} {dx}$ when $x=0$ using the following way: $=> \arccos y = m\arcsin x$ $=> - \dfrac {1} {\sqrt {1-y^2}} ...
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1answer
12 views

Solving a system of ODEs with variable in matrix A

I'm looking at a system x'=Ax as follows: $$\left[ \begin{matrix}x'(t) \\y'(t)\end{matrix} \right] = \left[\begin{matrix}0 & 1\\4/t^2 & -1/t\end{matrix}\right]\left[ \begin{matrix}x(t) ...
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8 views

Nonhomogeneous modified Bessel differential equation

I have a non homogeneous modified bessel equation of the form $$\dfrac{\partial^2 f}{\partial x^2}+\dfrac{1}{x}\dfrac{\partial f}{\partial x}-f=-AK_0(x),$$ for A is a constant which is determined ...
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1answer
9 views

Finding the eigenvalues and eigenvectors with each eigenvalue, solving the general solution with initial conditions.

Consider the system $x'_1 = x_1 + 2x_2$ and $x'_2 = 3x_1 + 2x_2$ If we write in matrix from as $X' = AX$, then a) $X =$ b) $X' =$ c) $A =$ d) Find the eigenvalues of A. e) Find eigenvectors ...
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10 views

Finding general solution of p(D)x

Reviewing for my math exam I came across a problem that I didn't know how to do and was unable to find any solutions online. The problem is: Find the general solution of $p(D)x = 4e^{3t} + 2\cos(t)$, ...
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13 views

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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13 views

Fuchs type equation [on hold]

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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12 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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10 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
22 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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26 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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26 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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22 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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27 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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17 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
21 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?? ...
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53 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
34 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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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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15 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 ...
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1answer
28 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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1answer
38 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
12 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
14 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
33 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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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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15 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
17 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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178 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
58 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
27 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
32 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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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 ...
2
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0answers
29 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 $n \times n$ ...