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 variables. For questions specifically concerning partial differential equations, use the [tag:pde] instead.

30,329 questions
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What can conclude from this two examples

so i stumbled across two task that says, is the given equation unique 1) yderived = sqrt(y) + 1 , y(1) = 0 So this equation is continues, but it does not satisfy Picard theorem, because dF/dy = 1/...
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Center, Stable and Unstable subspaces.

I'm trying to do an exercise in Perko's Differential equations and dynamical systems chapter 1 section 9 Stability theory that states the following. Let $A$ be a nonsingular $n\times n$ matrix and ...
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First order differential equation with y,y', and square root of y

I have been struggling with this equation: $(x^2+1)y'-2xy=4\sqrt{(x^2+1)y}\arctan x$ I have tried with $y=z^m$ to make homogeneous equation, but I didn't get anything anything useful. Left side also ...
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Solved inverse Galois problem for $\mathbb{C}(z)$ seems to contradict the theory about Liouvillian extensions.

The theory about Liouvillian extensions tells us that a Picard-Vessiot extension $L \supset k$ is Liouvillian if and only if the identity component $G^°$ of $G = Gal(L / k)$ is solvable. I think I ...
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Sufficient conditions for bounded output of a linear multi-step method

The system: \begin{align} \dot{x}(t) &= f(x(t))\\ x(t_0) &= x_0 \end{align} is being solved by a linear multi-step method (assume perfect initialization): \begin{equation} \sum_{j=0}^sa_j\...
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Tricky differential equation with $\sqrt{xy}$

I'm stuck with the following differential equation $$y' \sqrt{xy} - y - \sqrt{xy} + x = 0.$$ First I thought it's a Bernoulli equation but is isn't. I don\t have any further ideas. I would really ...
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Stochastic differential equation for random gaussian vibration

What would be the stochastic differential equation to obtain this equation $y(t) = \sum_{i = 1}^n A_i \sin(\omega_i t+\phi)$, where $A_i$, $\omega_i$ and $\phi_i$ are random variables and $t$ is time. ...
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Problem with making rigorous some arguments regarding a (stochastic) non-linear ordinary differential equation

I have a problem understanding chapter three of this paper by Ramirez et al. Let $b'_x$ denote white noise and $L^*$ the closure of smooth functions of compact support in $\mathbb{R}_+$ under the ...
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Proving that a function in an ODE has an asymptote

So I've looked at this answer to the problem of showing that the function $y$ that satisfies: $$y'=1+y^4$$ has an asymptote. The solution seems very elegant, except I cannot follow one of the steps. ...
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n linear independent set of n-times continuously differentiable function on an open interval forms a n-th order differential equation?

Let $\{\varphi_1(t),\varphi_2(t),\varphi_3(t),\dots,\varphi_n(t)\}$ be a linear independent set of $n$-times continuously differentiable functions on an open interval $I\subset\mathbb R$. How can I ...
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Two second order ODEs convert into a system of first order ODEs

$$2x_1''=-4x_1+3x_2$$ $$\frac{9}{4}x_2''=-\frac{27}{4}x_2+3x_1$$ I should these two second order ODEs as a system of first order ODEs. And their coefficient matrix should become $4$ by $4$ matrix. ...
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how to find Green function for boundary value problem

I know there's pretty generic algorithm, but I am stuck a bit. The initial problem is: $$y'' - y = f(x) \quad y'(0) = 0 \quad y(\pi) = 0$$ so I do: $$\lambda^2-1 =0 \quad \lambda_{1,2} \pm 1$$ ...
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Riccati equation (is my answer correct?)

I wanted to ask is my answer for solving Riccati equation correct. $(dy/dx) + (y^2/x^3) + (y/(2x)) + (x/2)$, i know that partial solution of the equation is $y = x^2$ so i need to find general ...
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Question regarding modeling Newton's Law of Cooling/Warming

A cup of coffee cools according to Newton’s law of cooling (see below). Use data from the graph of the temperature T(t) in Figure 1.3.9 to estimate the constants Tm, T0, and k in a model of the form ...
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Question regarding mathematical models and population

The population model given by: $$\frac{{dP}}{{dt}} = kP$$ (where k is a constant of proportionality.) fails to take death into consideration; the growth rate equals the birth rate. In another ...
Solve differential equation $u_t = i u_{xx} - x^2 u$
Consider $u_t = i u_{xx} - x^2 u$ with $u_{t = 0} = 1$. We want to find a solution. My attempt : let's say $u = X(x)T(t)$, hence we have $\frac{T'(t)}{T(t)} = i \frac{X"(x)}{X(x)} - x^2$. We may ...