# 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.

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### Uniqueness of the Solution of a Ordinary Differential Equation

Let $f$ a function, and the Cauchy problem $$x'=f(t,x) \qquad x(t_0)=x_0$$ I am studying EDO from two books, and I have a question about the uniqueness of the solution. Book 1: If $f$ is $C^1$ ...
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### find the solution of $2x\sin{\left(\frac{y}{x}\right)}dx+3y\cos{\left(\frac{y}{x}\right)}dx-3x\cos{\left(\frac{y}{x}\right)}dy=0$

A solution of the equation $$2x\sin{\left(\frac{y}{x}\right)}dx+3y\cos{\left(\frac{y}{x}\right)}dx-3x\cos{\left(\frac{y}{x}\right)}dy=0$$ I know the answer $c\sin(3y/x)$ but I don't know the solution
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### Find the general solution of $y''+p(t)y'+q(t)y = 1+t$

A solution of the equation $$y''+p(t)y'+q(t)y = 0$$ is $1+t^2$, and the Wronskian of any of two solutions of the equation is constant. Find the general solution of $$y''+p(t)y'+q(t)y = 1+t$$ I have ...
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### Approach for non linear ODE with quadratic derivative.

What is the approach to solve the following nonlinear ODE: $$\ddot{x}(t) + a(\dot{x}(t))^2+b\dot{x}(t)+kx(t) = 0$$ where $a$ is positive when $\dot{x}<0$ and negative otherwise. I am trying to ...
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### Stability of the trivial solution of a system of differential equations

I am trying to determine the stability properties of the equilbrium solution $(x,y) = (0,0)$ of the following system of ODEs: $$\dot x = x - y + kx(x^2+y^2), \\ \dot y = x - y + ky(x^2+y^2),$$ ...
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### Proof that the zero solution of equation $y''+f(y)=0$ is sustainable

I am trying to solve this prolem. Let $f(0) = 0$ and $tf(t)>0$ for $t \neq 0$. Proof that the zero solution of equation $$y''+f(y)=0$$ is sustainable. I thin it is a bad way to solve it by using ...
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### First order Non-Linear ODE with no explicit form of the derivative

I have the following non-linear first order ODE $$[a+b(1-e^{-m\frac{dy}{dx}})]\frac{dy}{dx}=f(x)$$ to be integrated over the range $x=x_0$ to $x=x_1$ The ODE is of the form $$y'=f(x,y,y')$$ ...
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### An explication about the derivate of poincaré maps

I am studying about derivate of poincaré maps $(\mathbb{R^2})$ but i don't know how to used this. I don't know how found the poincaré maps too. Someone can give me an easy exemple or open my mind in ...
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### What is the solution of this dif eqn [closed]

enter image description here The question is not in english but it is clear. Asking the solution with given initial conditions
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### How do you solve $(D^2+1)y = \ln|\cos(x)|$

I am stuch on this question: $$(D^2+1)y = \ln|\cos x|$$ where $D^2$ denotes the differential operator $d^2y/dx^2$ I suppose that I will begin with these lines: For cos(x) function use $(D^2+1)$...
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### Central Difference Approximations

Hi Guys I was going through the different approximations which can be used for differentiation such as the forward difference, the backward difference and lastly the central difference approximations. ...
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### Differential Equation using finite difference method

I am working on the following question $$y''+8(\sin^2 \pi y) y=0$$ where the initial conditions are $$y(0) = y(1) = 1$$ Now by the finite difference method i have made the substitution for $y''$ ...
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### Trivial Solution in Differential Equation

in the following differential equation: $xy' + y = y^{-2}$ we can see that $y=1$ is a solution that always satisfies the equation regardless of the value of $x$. Do we call this a trivial ...
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$y u_x-xu_y=0,u=g$ on $\Omega$ has a unique solution in neighborhood of $\Omega$ for every differentiable function g: $\Omega \rightarrow R$ if 1.$\Omega =\{(x,0):x>0\}$ 2.$\Omega =\{(x,y):x^... 1answer 19 views ### What does it mean when a system is made dimensionless and what is the exact technique for that? For school research I'm working on a system of ODEs to describe a chemical oscillator (the Oregonator). This system is described with the following system: $$\frac {dX}{dt}=k_1AY-k_2XY+k_3AX-2k_4X^... 0answers 4 views ### find the value of instablity error from which this value shows instability I have an Euler method that has this form:$$\hat{I}(t_{n+1}) = \hat{I}(t_{n})+h\beta \hat{I}(t_{n})(1-\frac{\hat {I}(t_{n})}{N})$$which can also be written like$$\hat{I}(t_{n+1})=\phi (\hat{I}(t_{... 0answers 16 views ### Is it always possible to get the direct function for a system of ODEs? Is is possible for all ODEs to derive the direct formula? I'm wondering if there is some (very difficult) mathematical method to get the direct formula. As an example, consider the following system: ... 1answer 26 views ### The integral$\int \frac{1}{\sqrt[y]{y}} dy$and the differential equation$y = \left(\frac{dy}{dx}\right)^y$I couldn't find a question about this integral, sorry if a similar question has been asked before. For fun, I came up with the differential equation: $$y = \frac{dy}{dx}^{{\frac{dy}{dx}}^{{\frac{dy}... 2answers 23 views ### Is dX/dt=X(t) the correct ODE for X(t)=e^t? For a school project for chemistry I use systems of ODEs to calculate the concentrations of specific chemicals over time. Now I am wondering if$$ \frac{dX}{dt} =X(t) $$the same is as$$ X(t)=e^... 0answers 16 views ### How to come up with Dulác-Function? I'm currently studying dynamical systems and came across the Bendixson-Dulac-Theorem Let$D \subseteq \mathbb{R}^2$open,$f \in \mathcal{C}^1(D, \mathbb{R}^2)$and consider the nonlinear system$...
The initial value problem $(x^2−x)\frac{dy}{dx}=(2x−1)y$, $y(x_0)=y_0$ has no solution if $(x_0,y_0)$ equals : Select one: A. $(1,1); \quad$ B. $(0,0); \quad$ C. $(2,1);\quad$ D. $(3,1)$.