# 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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### Finding the lower bound for Radius of Convergence of a Frobenius Series

this is my first post so I'll try to be brief. I am working on finding a problem such as this: given the ODE: $$(x^2-1)y''+x^2y' + cotx\cdot y = 0$$ "Find all the singular points of the following ...
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### A difference equation or functional equation that has a solution $(ax+b)^c$

I am looking for a (system of) functional equation or difference equation that has a solution: $f(x)=(ax+b)^c$, where $a,b,c$ are constants. Constant $a,b,c$ cannot appear in the equation. For example,...
18 views

### Find the difference equation given the general solution $y(k) = c_{1}5^{k} + c_{2}(-5)^{k} + c_{3}6^{k}$

Given that $y(k) = c_{1}5^{k} + c_{2}(-5)^{k} + c_{3}6^{k}$, is the general solution to a difference equation, how do you work backwards to find the difference equation?
1 vote
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### Decide whether given function is the general solution to ODE

I have a set of function and must decide whether each of them is a general solution to an ODE of the form : \begin{cases}y'(x)=f(x,y(x))\\y(x_0)=y_0\end{cases} When the an ODE comes obvious for a ...
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1 vote
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### Equality of functionals/relations upon interchanging arguments implying equality of functions?

Say we have a pair differentiable functions $Z(z)$ and $R(r)$ of the real variables $r,z$. We also have another pair of functions $\hat{Z}(z)$ and $\hat{R}(r)$ of the same variables. We have an ...
33 views

### Is there any theorem for the solutions $u(t)$, $v(t)$ of the following differential equation? [closed]

Suppose I have the following equation (where $a(t)$, $b(t)$ and $c(t)$ are continuous functions) $a(t)(u'(t))^2 + b(t)u'(t)v'(t) + c(t)(v'(t))^2 = 0$ Is there any theorem that could possibly tell me ...
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### Rewriting system of second order differential equation as system of first order

Given a charged particle moving in an electromagnetic field. We have $N$ amount of point charges placed in $\mathbb{R}^2$ on the coordinates $p_i$. We also have a free particle moving in $\mathbb{R}^2$...
1 vote
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### $x'(t)=|t-1|x(t)+1,x(0)=0$ Find $x(2)$ (using integrals is possible)

$x'(t)=|t-1|x(t)+1,x(0)=0$ Find $x(2)$ (using integrals is possible). First,how am I suppose to use $x(0)=0$ if I find the solution for $t>1$ and $t<1$ and I have to find $x(2)$ ? My solution : ...
1 vote
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### Singular perturbation differential equation

Find solutions for this system of differential equations with singular perturbation $$x'(t)=y, x(0)=x_0 \\ \epsilon y'(t)=\pm y, y(0)=y_0$$ with $t\geq 0, 0< \epsilon \ll 1$ for both cases ($\pm$). ...
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Given constants $C$, $v$ and $k$, consider the following ODE $$\dot u = C \left( v^2 - 2 u v + u^2 \right) - k$$ I have to find $u$. Because the equation is a first order non-linear differential ...
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### How many functions are possible for the given differential equation?

There was this question asked in a competitive examination , the solution of which is very confusing to me. The number of differentiable functions $y:(-\infty, \infty) \rightarrow [0, \infty)$ ...
1 vote
53 views

### Differential equation (encyme reaction)

Consider this enzyme reaction with initial conditions $c_1(0) = c_2(0) = p(0) = 0,s(0) = s_0, e(0) = e_0, e^{\ast}(0) = e^{\ast}_0.$ I determined this differential equations for the enzyme reactions: ...
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### how do i solve non linear equation? TOPIC IS PDE [closed]

Find a separated solution of the following nonlinear wave equation: ∂u/∂t=cu ∂y/∂x and What is a separated solution of the 2 -dimensional wave equation (∂^2 u)/(∂t^2 )=a (∂^2 u)/(∂x^2 )+b (∂^2 u)/(∂y^...
1 vote
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### An inequality for a maximal solution of an IVP [closed]

We have the function $f : \mathbb{R} \times \mathbb{R} \to \mathbb{R}, (x,y) \mapsto \frac{xy}{\sqrt{y^2+1} }$ and the following IVP \begin{align*} y'=f(x,y), \qquad y(0)=1. \end{align*} How does one ...
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### Solution of 2nd order ODE [closed]

How can I solve the following ordinary differential equation (ODE) analytically? $$a(u)\frac{d^2u}{dx^2}+\left(\frac{du}{dx}\right)^2 \frac{da(u)}{du}+b =0$$
1 vote
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### O.D.E. with elliptic integral

I have an energy integral $W=\int V dx$, where after using spherical parametrization $V$ is $V=\frac{1}{2}[(\theta'-\lambda)^2+\gamma^2\cos^2\theta]$. Stationary points of the equation, using Euler -...
1 vote
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### Euler integration solution from system of ODE's - already estimated values

I am currently completing an investigation assignment on modelling the growth of a virus inside of the host. There are 3 ODEs that I am using in the system, all determined by change in t. The ...
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### A basic question on the introduction of contracting system

I am learning contraction theory from this tutorial which starts by calculating the difference between two arbitrary trajectories of a scalar system (corresponds to 26:30 in the video): the ...
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### fourth-order finite difference for $(a(x)u'(x))'$

Previously I asked here about constructing a symmetric matrix for doing finite difference for $(a(x)u'(x))'$ where the (diffusion) coefficient $a(x)$ is spatially varying. The answer provided there ...