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

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9 views

Undetermined coefficients when RHS is a polynomial

When applying undetermined coefficients to a non-homogeneous linear differential equation, we let $y_p=A_0+A_1x+\cdots A_n x^n$ when a degree $n$ polynomial is on the RHS. Doesn't this potentially ...
2
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1answer
21 views

Solve $(2x - 1)y'' - 4xy' + 4y = 0$

I am self-studying differential equations using MIT's publicly available materials. One problem set question asks us to first verify that $x$ is a solution to $(2x - 1)y'' - 4xy' + 4y - 0$ (with $2x ...
1
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0answers
14 views

Differential equation where Picard-Lindelöf can not be applied

My question is the following : Let $f:\mathbb{R}\to\mathbb{R}$ be continuous function and let $u:[a,b]\to\mathbb{R}$ be a $C^1$ function such that $$\forall t\in[a,b],u'(t)=f(u(t))\text{ and ...
1
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2answers
24 views

How to solve the differential equation $y' + \sec(x)*y = \tan(x)$

I am really struggling to solve the differential equation: $y' + \sec(x)y = \tan(x)$. If someone could point me in the right direction or give me a step by step plan it would be much appreciated! So ...
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2answers
23 views

Linear Stability Analysis of an ODE

Question: Find the equilibrium points of the following ODE and then use Linear Stability analysis to determine the stability. $$\frac{dy}{dt}= cy - \frac{y}{1+y^2}$$ My Attempt: I have worked out ...
1
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1answer
10 views

Confusion regarding the Cauchy-Euler equation

In my textbook it says that: An equation of the form $$a_2 x^2 \frac{\mathrm{d}^2y}{\mathrm{d}x^2}+a_1 x \frac{\mathrm{d}y}{\mathrm{d}x}+a_0 y=f(x)$$ is called ...
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2answers
16 views

Lyapunov Exponent sensitivity to initial conditions

I am plotting the Lyapunov exponent as a function of a parameter $r$ with an initial condition $x_0$. The equation looks like this: $$x_{n+1} =4rx_n (1-x_n)$$ When I try different initial conditions ...
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2answers
26 views

Find the general solution of $y'= a^{x + y}$ where y is the function

How should I approach this problem? Should I put ln logarithm on both sides and move factor of variable of lnx on right side in front of the ln, or do somehing else? I understand general solving and ...
0
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1answer
12 views

On the computation of the Hessian matrix.

I'm trying to compute the Hessian matrix of a data fit of an ODE model to some data. Below is a cut out of the instructions I'm following (which can also be found at ...
0
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1answer
19 views

Proving uniqueness using $\dfrac{\partial}{\partial y}$?

I remember in the beginning of my undergrad linear differential equations class (while or before we were introduced to linear ODE's), we proved the uniqueness of a solution to an IVP by taking the ...
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0answers
26 views

Solving Secondary Linear ODE : $xy''+y'-xy=0$

The ODE is$$x\frac{d^2y}{dx^2}+\frac{dy}{dx}-xy=0$$ I thought that this equation looks very similar with Bessel's equation so I tried substitution $t=ix$. Then by $\frac{dy}{dx}=i\frac{dy}{dt}$ and ...
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1answer
17 views

differential equation with non differentiable non homogeneous part [on hold]

i am not able to solve this please if somebody could help
0
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1answer
11 views

Determining the stability of a system using Lyapunov function

Consider the nonlinear dynamical system describing motion of a simple pendulum with viscous damping given by $$\ddot{\theta}(t)+\dot{\theta}(t)+gl\sin(\theta(t))=0,\quad \theta(0)=\theta_0, ...
3
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1answer
91 views

Why is “$\pi^2= g $” where $g$ is the gravitational constant?

Some months ago a professor of mine showed us a 'proof' of why $g\approx 9.8 ~\text{m}/\text{s}^2$ (the gravitational acceleration at the surface of the Earth) is 'equal' to $\pi^2\approx9.86\dots$ ...
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1answer
23 views

To find $b$ and $c$ such that solution of differential equation goes to $0$.

I tried by writing the quadratic formula and think the answer is $B$, however I am not sure.
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0answers
11 views

Analytical solution of parabolic equation

What is the analytical expression of the solution of the following 1D parabolic equation ? $$\dfrac{∂f}{∂t}+V\dfrac{∂f}{∂x}=abe^{at}$$ where $t, x$ – independent variable (time and space position ...
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0answers
12 views

Exercise: ODE with specified initial condition.

a)Consider the following ODE: $y'=4t \sqrt y$ $y(0)=1$ and find the unique solution. b)What is required for a numerical method to solve the problem exactly? c)Now consider the modified problem: ...
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0answers
23 views

Solving a general Bernoulli diferential equation.

So this is the equation. $a,b,N_0 \in \mathbb{R}, ab>0$ $$N'(x)= aN(x)-bN(x)^2, (x\in[0,\infty)), N(0) = N_0$$ First things first $\color{red}I $ need to transform this, If we let $N = w^{1/2}$, ...
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4answers
47 views

Solving $-\epsilon u''(x) + \beta u'(x) = 1$

Consider the equation: $$-\epsilon u''(x) + \beta u'(x) = 1, \;\; x \in (0,1)$$ $$u(0) = 0, \;\; u(1) = 0.$$ $\beta > 0, \;\; \epsilon > 0$. Can someone please help me to solve this equation? ...
0
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0answers
68 views

If $y(t) = t\left(1-\int_0^ty(x)\,dx\right)+4\int_0^tx\,y(x)\,dx,$ then $\int_0^{\pi/2} y(t)\,dt$ is equal to?

Leibniz rule or Laplace transform? Let $y(t)$ be a continuous function on $[0,\infty)$. If $$y(t) = t\left(1-4\int_0^ty(x)\,dx\right)+4\int_0^tx\,y(x)\,dx,$$ then $\int_0^{\pi/2} y(t)\,dt$ ...
1
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2answers
31 views

Analytic function $f,$ such that $f(0) = 1$ and $f'(z) = zf(z),$ for all $z \in \mathbb{C}$

I'm trying to find an example of an analytic function $f$ satisfying the IVP $$ f'(z) = z\,f(z), \quad f(0) = 1, $$ and for all $z \in \mathbb{C}$, but I'm somewhat at a loss of the best way to ...
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0answers
22 views

How are the parabolic cylindric functions to be used?

It may seem a weird question but the documentation about them is quite scarce. I will refer to the equations explained at this page: https://archive.lib.msu.edu/crcmath/math/math/p/p058.htm $$ y'' + ...
2
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1answer
74 views

Differential equation: $\ddot{y}(x) + \alpha\dot{y}^2(x) + \beta y(x) = 0$

I am interested in finding an approximate solution for this differential equation, since the exact analytic solution seems to not exist. I tried with Mathematica and it spits out nothing. ...
1
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3answers
41 views

Solving $yy''=(y')^2-y'$

I want to solve the ODE $yy''=(y')^2-y'$ with the initial conditions $y(0)=1, y'(0)=2$. My attempt: $$yy''=(y')^2-y'$$ $$(\frac {y'}y)'=(\frac 1y)'$$ $$\frac {y'}y=\frac 1y+c$$ This holds for all ...
0
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1answer
19 views

State transformation for non-holonomic differential equation.

Given a non-holonomic dynamical system, \begin{align*} \dot x = v\cos\theta \\ \dot y = v\sin\theta \\ \dot \theta = \omega \end{align*} with constraints $|v| < v_{max}, |\omega| < ...
2
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0answers
26 views

Finding when fixed point is hyperbolic

Consider the IVP for the $2$-dimensional dynamical system ($X=[0, \infty )^2$) $$\dot{x_1}=a-x_1-\frac{4x_1x_2}{1+x_1^2}$$ $$\dot{x_2}=bx_1 \bigg( 1- \frac{x_2}{1+x_1^2} \bigg)$$ for all $t \in I$, ...
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0answers
13 views

Critical point of an ODE

I have been asked to deduce if an ODE has a critical point from drawing its isoclines and then sketching the integral curve. What exactly is a critical point of an ODE, and how would I deduce it from ...
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0answers
17 views

Help following a published calculation of a prediction confidence interval for a prediction made from a simple ODE

I'm trying to follow a calculation made in a paper(section 2 from the supplementary contents of ...
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0answers
15 views

Finding a Lyapunov function for system of non-linear differential equations

I have the following system of differential equations and I would like to find a Lyapunov function for it, in order to classify the point $(0,0,0)$: $$ \begin{cases} x' &= -y-xy^2+z^2-x^3,\\ y' ...
0
votes
1answer
27 views

Differential equation: $e^{xy^2}\frac{x}{x^2+1}\,dx - e^{y^2x}\,dy = 0$

I have no idea how to solve this and would love some help. $$e^{xy^2}\frac{x}{x^2+1}\,dx - e^{y^2x}\,dy = 0$$
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2answers
29 views

Vector norm and relationship with euclidean distance

If $y\in E_n$ (n dimensional euclidean space) show that $||\textbf{y}||\leq|\textbf{y}|\leq \sqrt{n}||\textbf{y}||$ Where $||\textbf{y}||$ is the euclidean length of the vector $\textbf{y}$ and ...
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3answers
42 views

What exactly is the maximal solution of an ODE and why do we care?

I am reading these notes on the definition of a maximal solution of an ODE i.e. http://www.math.lmu.de/~philip/publications/lectureNotes/ODE.pdf But the definition is so abstract and no example is ...
2
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0answers
11 views

Finding a lower and upper bound for the first eigenvalue of a Sturm Liouville problem

Given the eigenvalue problem $y''+\lambda (1+x^2)y=0$, $y(0)=y(1)=0$, I need to find a lower and upper bound for the first eigenvalue $\lambda_0$ (that fits the eigenfunction that has no zeros in ...
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1answer
9 views

Understanding the steps taken in a calculation of the maximum profile likelihood of a simple ODE, given some data

I'm trying to understand a calculation made in a paper (section 2 from the supplementary contents of ...
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0answers
26 views

Is there a close-form solution for the non-linear difference equation? [on hold]

is there a close-form solution for the difference equation below? $$(x_{n+2}-x_{n+1})-(x_{n+1}-x_n)=(\frac{x_{n+1}}{c})(x_{n+2}-x_n)$$ Any comments are appreciated.
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0answers
13 views

Dermining stable and unstable manifolds - is my result ok?

Determine all stable and unstable manifolds of the equilibria of $$ \dot{x}=x(1+x)(1-x). $$ Are there homoclinic/ heteroclinic solutions? Hey, just would like to know if I am ...
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0answers
12 views

Quasi-linear partial differential equations. Solving them.

This is what I have as a quasi-linear partial differential equation:$$u(x_1,...,x_n), \ \ \ \ \sum_{i=1}^{n}A_i(X,u) \frac{\partial u}{\partial x_i}=A_{n+1}(X,u) \ \ \ (1)$$ Then it says let ...
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2answers
36 views

How can I solve a first order ODE with $\pm$ signs by the Integrating Factor method?

I have the following first order ODE to be solved via the integrating factor method: $$\frac{\mathrm{d}z}{\mathrm{d}y}\pm z=-\frac12y\tag{1}$$ This is in the general form: ...
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0answers
19 views

Methods for first order PDEs in higher dimensions

What are the possible known methods for solving first order PDEs in higher dimensions? Is there anything else besides the method of characteristic curves? In particular, I have four first order, ...
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1answer
14 views

Show exponential stability quadratic form

Please help me with the following proof: Suppose $\dot x=f(x(t))$ and suppose that we have: $$ \frac{d}{dt}\left( x(t)^TPx(t) \right)\le -x(t)^TQx(t) $$ where $P$ and $Q$ are symmetric ...
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0answers
27 views

If a and b are negative , then can we use the same method we are taught for solving the equation y'' + ay' + by=0 ,

If $a$ and $b$ are negative , then can we use the same method that we are taught for solving the ODE which is $$y'' + ay' + by=0$$
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2answers
85 views

The solution of ODE $k'(x) = r(k(x))$ is infinitely differentiable if $r$ is

If there is a function $r(x)$ that is infinitely differentiable, prove that $k(x)$ is also infinitely differentiable if $k'(x) = r(k(x))$ for all $x$. I am trying to somehow prove using ...
1
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1answer
19 views

Solving Laguerre coefficients with Integral?

I'm having some difficulty understanding the solution to a particular Laguerre expansion. The problem reads "Expand the term $ e^{-x}$ as a Laguerre expansion, noting the orthogonality of $$ < ...
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0answers
36 views

Transient Terms in a General Solution

Find the general solution of the given differential equation: $$ (x^2-4)(\frac{dy}{dx}) +4y = (x+2)^2 $$ I found the general solution of the D.E and I got the following correct solution: $$ y = ...
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1answer
27 views

PDE Proof that a linear combination of 2 solutions is also a solution [on hold]

Can someone please help? I've been trying to figure this for a few days now. Consider the first order PDE: $au_t + bu_x$ = 0, where a and b are constants. Show that if $u_1$ and $u_2$ are solutions ...
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0answers
17 views

How to integrate to solve a PDE with mixed partials in the integrand

Problem Statement: Determine the equlibrium temperature distribution inside a circular annulus $r_1\leq r \leq r_2$. If the outer radius is at temperature $T_2$ and inner radius at temp $T_1$. So ...
0
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0answers
18 views

What is the largest t-interval on which guarantees a unique solution? [on hold]

What is the largest t-interval on which guarantees a unique solution for this equation? $$y'' + y'+ 3ty = \tan t,\quad y(\pi) = 1,\quad y'(\pi) = -1$$
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0answers
28 views

Solving $\vec y'=\begin{pmatrix} 1 & -1 \\ 0 & 1\\ \end{pmatrix}\vec y$

I need to solve $\vec y'=\begin{pmatrix} 1 & -1 \\ 0 & 1\\ \end{pmatrix}\vec y$. The characteristic polynomial is $(r-1)^2$, so the only eingenvalue is $1$. I found ...
0
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1answer
28 views

Showing that if $y(x)$ is a solution, then $y(-x)$ is also a solution for a specific ODE

Given the ODE $(1-x^2)y''-xy'+\alpha^2 y=0$, I need to show that if $y(x)$ is a solution, then $y(-x)$ is also a solution. From what I understand, because $y(0)=y(-0)$, it means that all solutions are ...
1
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0answers
35 views

Different answer when using the 'method of undetermined coefficients' compared to Laplace transform

I have an ordinary differential equation: $$ \frac{\mathrm{d}^2u}{\mathrm{d}t^2} + u = \mathrm{e}^{-t}\cos(t)$$ with $u(0) = u_0$ and $\dot{u}(0) = v_0$, when using the method of undetermined ...