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

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0
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1answer
96 views

Inverse laplace transform of a hard partial fraction, $1/[s^2(s^2+\omega^2)]$

So the question is find the inverse of $\dfrac{1}{s^2(s^2+\omega^2)}$. And here is the solution. I have no idea why its done this way. I would think to take a partial fraction of the form ...
1
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2answers
104 views

Asymptotic stability of the zero solution of the equation $x'=x/(1+t)-x^3$

Problem Prove the zero solution of the following equation is asymptotically stable $$\frac{dx}{dt}=\frac{x}{1+t}-x^3$$ Progress The equation is nonlinear and non-autonomous. Without the nonlinear ...
1
vote
1answer
110 views

Inverse Laplace transform of $(2s+4)/(s^2+4s+5)^2$

Out of many transformation shortcuts in Laplace table I still find difficulty in finding the inverse laplace transform of $\displaystyle \frac{2s+4}{(s^2+4s+5)^2}$. I tried partial fraction and its ...
0
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0answers
113 views

Generalized Leibniz Rule

Leibniz Rule states that, $$(f\cdot g)^{(m)}(x)=\sum_{k=0}^m \binom{m}{k} f^{(m-k)}(x)g^{(k)}(x).$$ Writing this with differentiation denoted by $D$, we might say $$D^m (fg) = \sum_{k=0}^m ...
1
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1answer
275 views

Finding the solution $u(x,y)$ to Laplace's equation in a rectangle.

I have the following problem which I basically understand, but I cannot understand how my professor did a substitution almost at the end of the problem. Thanks a lot in advance! Question: Find the ...
0
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0answers
93 views

Implementing formula correctly in matlab for neuroscience: total soma membrane potential?

Please help me to understand: am I correctly implementing a total soma membrane potential (TSMP) equation in Matlab? Due to being a new member I need to use this list link to refer to the links ...
3
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3answers
42 views

Solving some inhomogeneous differential equations

I am currently reviewing some differential equations and ran into a couple of problems with the problems shown below particularly in the form of the particular solution for the equations. I haven't ...
1
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0answers
44 views

Equation of a curve with a local minimum fixed at $x=a$ when we rotate the curve about the origin.

We have a strangely curved plank. If we place a round weighted object on it, it will rest itself at one point of it, when we incline the plank slowly, the object will gradually move towards a ...
4
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0answers
95 views

$u''+\frac{4}{x+1}u'+\frac{2}{\left(x+1\right)^{2}}u=0$ variational solution

This is a concept solution scheme derived from a particular example that I have not been able to generalise sufficiently. The objective is to find a particular solution to a certain second-order ...
0
votes
1answer
47 views

equilibrium point of an ODE

If a trajectory of the ODE $x'(t) = f(x(t))$ reaches the equilibrium of it then is it true that it stays there forever ? Why ? I think I am missing something trivial (Hint enough).
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3answers
111 views

A solution for a system of differential equations?

I want to check answer for specific ODE solvers, for instances, solving: $x_1' = 1/5\; x_1 + 4/5\; x_2$ $x_2' = 4/5\;x_1 + 1/5\; x_2$ $x_1(0) = 1$, $x_2(0) = 3$ I've just learnt how to solve these ...
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0answers
50 views

Continuity of the solution on the initial condition

Let $F \colon \mathbb{R}^d \to \mathbb{R}^d$ a continuous function and assume that the ODE $x' = F(x)$ with initial condition $x(0)=u$ has a unique solution $\phi(t,u)$ defined for all $t\in ...
1
vote
2answers
130 views

Judicious guess for the solution of differential equation $y''-2y'+5y=2(\cos t)^2 e^t$

I want to find the solutions of the differential equation: $y''-2y'+5y=2(\cos t)^2 e^t$. I want to do this with the judicious guessing method and therefore I want to write the right part of the ...
0
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1answer
88 views

Two methods of finding a function $f$ such that $Mdx+Ndy=0$ on the curves $f(x,y)=c$

this problem is from my class,i did one way and got one answer,professor did it in another way and got another answer.question is:Find $f(x,y)=constant$ where differential equation is ...
0
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2answers
41 views

Find the general solution to

Find the general solution to: $$\frac{d}{dx}\left( x^3\frac{du}{dx}\right)=0.$$ What I did was integrate twice and I got $$u(x)= c_2 - \frac{c_1}{2x^2}.$$ But the answer is $$u(x)=c_1 + ...
1
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0answers
116 views

Central manifold theorem => Stable/unstable manifold?

I'm a bit confused why we always separate the stable/unstable manifold theorem and the central manifold theorem. The stable/unstable manifold theorem applies to a hyperbolic point ...
0
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1answer
49 views

Using the Laplace transform to solve an ODE with piecewise input

I have the answer to this problem. My question is with the function $u(t)$. $u(t)$ is: $$u(t) = 2\cos(t)+2\sin(t-\pi/2)*1(t-\pi/2)$$ Why is there a $1(t-\pi/2)$ multiplying the $2\sin(t-\pi/2)$? ...
1
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3answers
112 views

Eigenvalue products

Prove that if $detA > 1$ then $A$ has at least one eigenvalue with $|\lambda |> 1$. The answer says: If all $|\lambda_j | \le 1$ then so is their product $1 \ge |\lambda_1 ...\lambda_n| ...
0
votes
1answer
57 views

Local isometric embedding

Every $n$-dimensional smooth Riemannian manifold admits a local isometric embedding of class $C^1$ into $\mathbb R^{n+1}$ by the Nash-Kuiper theorem. An example by Nadirashvili and Yuan shows that in ...
1
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2answers
67 views

Finding solution basis of $y^{(4)}-2y'''+5y''-8y'+4y=0$

Find a real-valued solution basis of $$y^{(4)}-2y'''+5y''-8y'+4y=0.$$ The corresponding characteristic equation is $$x^4-2x^3+5x^2-8x+4=0$$ $$\iff(x-1)^2(x^2+4)=0$$ which has the zeros $1, 2i, -2i$. ...
2
votes
2answers
151 views

1st order separable ODE involving the complex conjugate of the dependent variable

Is there a closed form (complex) solution $z(t)$ to the equation \begin{align} \frac{dz}{dt}=f(t)\bar{z}, \end{align} (the bar means complex conjugate) for any given complex valued function $f$ of a ...
3
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1answer
3k views

How to solve partial integro-differential equation?

Suppose the following partial integro-differential equation for a function $u(x,t)$ with $t\geq0$, $x \in [0,L]$: $\partial_t u = \partial_{xx} u + f(u,\lambda)$ $\lambda = B\left(u_0 - \int_{x=0}^L ...
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0answers
21 views

How to find error constant, global and local error of ODE integration method

In literature regarding open channel flows I bumped into strange ODE integration methods: the first one: $$y_{i+1}=y_i+\Delta x\cdot\sqrt{f_i\cdot f_{i+1}}$$ the second one $$y_{i+1}=y_i+\Delta ...
1
vote
1answer
33 views

How to normalise equations of the form $dy/dx=B$ and $d^2y/dx^2=A$?

So I am trying to normalise equations of the form, $$dy/dx=B \mbox{ and } d^{2}y/dx^{2}=A$$ If I define $y^{*}$ as; $$y^{*}=By \Rightarrow dy^{*}/dy=B $$ Is it also then true that, $$d(dy^{*})/dy = B ...
2
votes
3answers
199 views

How to reduce this to Sturm-Liouville form?

I have the ODE $$-(1-x^2) \frac{d^2 f(x)}{dx^2} + x \frac{df(x)}{dx}+g(x)f(x)=\lambda f(x)$$ and I want to reduce it to Sturm-liouville form. The problem is that we don't have $2x$ but just $x$. ...
2
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0answers
44 views

Special properties of bounded functions

I have a problem understanding the reasons as to why under some circumstances a term can be omitted due to it being a part of a bounded function, and I hoped to get some clarity to this here. There is ...
1
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2answers
62 views

Differential Equation with sketching graph

I'm trying to solve the equation: $$ y' = \frac{x\left(x^2+1\right)}{4y^3},\,\,y(0) = -\frac{1}{\sqrt{2}}\tag{1} $$ a) Find explicit solution and determine the interval in which the solution is ...
2
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2answers
135 views

Stable manifold for bidimensional nonlinear dynamic system with complex eigenvalues

Given a autonomous nonlinear dynamic system of the form $$f(x,y)=\begin{bmatrix} B_1 x + g_1(x,y) \\ B_2 y + g_2(x,y) \end{bmatrix}$$ with $B_i\in\mathbb{R}$ (bidimensional problem), with $g_i$ ...
0
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0answers
32 views

Representative solutions of differential equations

Consider the equation $x''-x^2=-25$. How can we sketch some representative solutions? If we make this equation to a first order system and use linearization would that be a good approach? Also, how ...
2
votes
3answers
94 views

Bessel function ratio approximation

Can we say anything about the ratio: $$\frac{K_1(z)}{K_0(z)}?$$ In particular, can we describe its behaviour for small or large $z\in\mathbb{R}$.
0
votes
1answer
33 views

Finding the constant for the particular solution to $y''(x) + y(x) = 2^x$

I'm really confused on this problem. Right now, I'm solving for the particular solution of: $y''(x) + y'(x) = 2^x$ My test solution was $A^x$, and I got $yp(x) = 2^x/(ln(A)^2 + 1)$ My problem here ...
0
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0answers
79 views

Relationship between $\sum\limits_{n=0}^\infty \frac{a_n x^n}{n!}$ and $\sum\limits_{n=0}^\infty \frac{a_n^2 x^n}{n!}$

For an analytic function with the property $f^{(n)}(0)=a_n$, we have $f(x)=\sum\limits_{n=0}^\infty \frac{a_n x^n}{n!}$. This can be extended to $f^{(n)}(x)=\sum\limits_{n=0}^\infty \frac{a_{n+1} ...
1
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1answer
33 views

Unable to get a particular solution for a system of ODE equations with the method of undetermined coefficients

so I have solved this problem using another method (the method of diagonalisation), but I now want to try with the method of undetermined coefficients and cannot get the right result for $\vec{b}$. ...
0
votes
1answer
75 views

What functions have vanishing boundary conditions for $0^{th}$ and $1^{st}$ order derivatives?

What (non-trivial) functions satisfy $f(a)=f(b)=f'(a)=f'(b)=0$? I am working on some undergrad research and would like to understand this a bit more. Context I am investigating a differential ...
2
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2answers
134 views

Intuition behind convolution identity for Laplace transforms

Convolutions, relatively speaking, are fairly straightforward for simple systems (from an applied perspective), but I cannot, at all, find the intuition behind the Laplace identity for convolutions. ...
3
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3answers
121 views

Differential equation with $\sqrt{xy}$

$$7\sqrt{xy} \frac{dy}{dx}=4, \quad x,y>0$$ How do I solve this equation for $y$
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0answers
67 views

Global existence without solving a differential equation

I have the following cauchy problem: $$ y(t)' = (y(t) - t^3 - 1)^3 $$ $$ y(0) = \alpha $$ Discuss the global existence when $1)\,\, \alpha < 1 $ $2) \,\,\alpha = 1 $ I tried the following: If ...
1
vote
2answers
69 views

Solve the following Ordinary Differential Equation

Suppose if we want to solve $\frac{d}{dx}u + f(x)u(x) = 0,$ then the solution is $u(x)=u(0)e^{-\int_0^xf(y)dy}$. Similarly what is the solution of $\frac{d^2}{dx^2}u- \frac{d}{dx}u - f(x)u(x) = 0,$ ...
1
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2answers
249 views

#26, the Inversion of Sugar

I'm trying to solve #26 from Chapter 7, Transcendental Functions (Thomas' Calculus 12th Edition) and I can't seem to figure out this problem: The Processing of raw sugar has a step called ...
0
votes
1answer
43 views

Partial fractions where the denominator is one function

I need to solve this differential equation for x: $$ \frac{dv}{dx} = \frac{4000}{v} - 0.9v $$ Rearranging: $$ \frac{dx}{dv} = \frac{1}{4000v^{-1} - 0.9v} $$ How would I go about splitting this ...
0
votes
1answer
104 views

Inhomogeneous Initial value problem.

The solution to the initial value problem $$x'(t)=Ax(t)+g(t)\quad\text{with}\quad x(0)=x_0$$ is $$x(t)=\exp(tA)x_0+\int_0^t \exp((t-s)A)g(s)\,\mathrm{d}s$$ Suppose that all eigenvalues of $A$, ...
0
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0answers
40 views

Stability of fixed points

Given $dx/dt=f(x)=(1+x)(b^2-x^2)$. If it has two unstable and one stable fixed points, then $b<-1$. To prove this I started with the fixed points $ x_{1,2,3}=-1,\pm b$. For stable point, ...
1
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0answers
59 views

Stochastic differential equation for a Fokker-Planck-type equation with a non-derivative term

I have something similar to a Fokker-Planck equation of the form $\frac{\partial}{\partial t}f( x,t) = A(x,t)f(x,t)- \frac{\partial}{\partial x}[B(x,t) f(x,t)] +\frac{1}{2}\frac{\partial ^2}{\partial ...
7
votes
1answer
132 views

Derivation of Schrödinger's equation

I recall a famous quote of the late physicist Richard Feynman: Where did we get that from? It's not possible to derive it from anything you know. It came out of the mind of Schrödinger. This ...
2
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0answers
130 views

How to solve a 4x4 linear system using a set of initial conditions

I have a non linear 4x4 system of ODEs. I linearized it about an equilibrium point (i am using floquet theory and i need the linearized system for that) and below is the linearized system: ...
0
votes
1answer
35 views

The meaning of predictor $u_{n+1}$ in the improved Euler's method

I have $$y\prime=9x, h=\frac{1}{2}, y(0)=1$$ And I want to find y(2) using the improved euler's method. And I know that it goes: $$y_{n+1}=y_n+h\frac{f(x_n,y_n)+f(x_{n+1},u_{n+1})}{2}$$ What I ...
2
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0answers
32 views

Growth condition in differential equation and vanishing solution at boundaries

In a discussion on solving a partial differential equation I lately read: "Under a standard growth condition on the solution at infinity, the resulting PDE is fully specified without boundary ...
3
votes
1answer
109 views

How to use the Mehler kernel to get the solution of the Quantum harmonic oscillator with a given initial condition

In this wiki-article http://en.wikipedia.org/wiki/Mehler_kernel the fundamental solution of the differential equation for the Quantum harmonic oscillator is provided by the Mehler Kernel: ...
0
votes
1answer
32 views

Partial differentiable equation

Let $f$ be a differentiable function of one variable and $u=\frac{x}{y}f(\frac{y}{x})$. If u satisfies a partial differential equation of the form $\frac{x}{y}u_{x} + u_{y}= G$, then who is $G$. I ...
2
votes
0answers
164 views

What Happens At An Equilibrium Point For An Autonomous First-Order Differential Equation.

Let $\frac{dx}{dt} =f(x)$ be an autonomous first-order differential equation with equilibrium point at $x_0$. a) Suppose $f'(x_0) = 0$. What can you say about the behaviour of the solution near ...