Questions tagged [ordinary-differential-equations]
For questions about ordinary differential equations, which are differential equations involving ordinary derivatives of one or more dependent variables with respect to a single independent variable. For questions specifically concerning partial differential equations, use the [tag:pde] instead.
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Second order nonlinear differential equation with missing first derivative.
I've got this nonlinear differential equation while solving the Navier-Stockes equation for a point source located at the apex of a corner:
$y^2 + 4\nu y + \nu y'' + C = 0$, where $\nu$, $C$ are ...
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Degenerate perturbation theory to nonlinear equation
I want to use perturbation theory to find the solution to the following nonlinear equation:
$$x_i\left(\sum_{j=1}^Nx_j^2\right)-a x_i + \epsilon \sum_{j\neq i}^N J_{ij}x_j=0,$$where $i=1\cdots N$ and $...
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Asymptotic differential equation
This is probably a stupid question, but suppose $x f'\sim -f$ in the sense that $\lim_{x\to \infty} \frac{x f'(x)}{f(x)} = -1$.
The solution to $xf' = -f$ is of the form $f(x) = Cx^{-1},$ so it is ...
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Change of variables by scaling.
I'm going through the book Applied Partial Differential Equations: with Fourier Series and Boundary value problems by Richard Haberman. In chapter 7 this is said about this equation (Next step is ...
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Ordinary differential equation of first order, but not of first degree [closed]
I am trying to solve, $$x=py+p^2,\text{ where } p=\frac{dy}{dx}$$ I solved it by derivating w.r.t. y, which is the same as done in my book
But if, while finding integrating factor, I solve it as, $$I....
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How would I prove or disprove this statement?
The question is to prove that $\forall n \in \mathbb{N}$ the differential equation $\prod_{i=0}^{n}{y^{(i)}}=\sum_{i=0}^{n}{y^{(i)}}$ has a solution (not including $y=0$) with a domain of $\Bbb{R}$ ...
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The homoclinic solution of the whirling pendulum equation [closed]
How to get the homoclinic solution of the whirling pendulum equation
$$
\left.\begin{array}{rcl}
&&
\\
x' & = & y
\\[2mm]
y' & = & \sin\left(x\right)
\left[\,\cos\left(x\right) ...
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Asymptotic stability and Lyapunov functions
I fail to understand a passage in the proof of the following theorem (right after the definition that gives the context of my question):
(Definition of Lyapunov function)
Let $\Omega$ be a ...
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Approximating solution to vector recurrence relation with element-wise exponential
$$
\mbox{Let}\ A \in \mathbb{R}^{n \times n}\ \mbox{and}\ \gamma_{1}, \ldots, \gamma_{n} \in \mathbb{R}\ \mbox{with}\ \gamma_{i} > 0,\ \forall\ i.
$$
$$
\mbox{Define}\ \operatorname{f}: \mathbb{R}^{...
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Stochastic differential equations with only time integrals
I want to reason about the stochastic differential equation
$$ dX_t = A_t X_t dt $$
Where $A_t$ is a matrix valued stochastic process, and hence $X_t$ is a vector valued stochastic process. Are there ...
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Does Solution to Banach Space ODE $\dot{x} = Ax$ with Non-Positive Spectrum Converges to Kernel?
Let $A:X \rightarrow X$ be a bounded linear Banach space operator. In this book, Daleckii and Krein investigate the behavior of the initial value problem $\dot{x} = Ax$ with $x(0) = x_0$ in terms of ...
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Is it possible to reverse construct a first-order homogeneous (no forcing term) difference equation from its general solution?
The solution to $y' = a(t)y$ is $y = Ce^{\int a(t)dt}$. If I start with a general solution to this ODE such as $y = Ct$, then I can use algebra to find that $a(t) = \frac{1}{t}$, and reverse ...
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The space of solutions for an ODE on an interval containing a regular singular point
Consider the differential operator $L(y)=x^ny^{(n)}+a_1(x)x^{n-1}y^{(n-1)}+\cdots+a_n(x)x^0y^{(0)}$, where each of $a_i(x)$ has a power series expansion at $x=0$ converging for all $|x|<r_0$ for ...
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Converting a 2nd order homogeneous equation into a known form of an Special function.
I got the following 2nd order fuchsian equation.
$z^2 u''(z)+z u'(z)+\left(\frac{A \beta }{(z-1)^2}+\frac{A (\alpha +\beta )}{z-1}-A
E\right)u(z)=0$
I am pretty impressed by how close it looks to ...
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Can we convert the following integral equation to a differential equation:$h(r)= \int_0^\infty\frac{f(x)}{e^{r x} + 1} dx$?
Can we convert the following integral equation to a differential equation:$$h(r) =\int_0^\infty\frac{f(x)}{e^{r x} + 1 } dx?$$ Here, $f(x)$ is a non-trivial 'nice' function( whatsoever condition is ...
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