# 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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### Qualitative theory of systems of ODE that involve thousands of functions/equations?

I am trying to learn systems biology modelling (https://arxiv.org/pdf/1711.08079.pdf is example article that handle the parameter identification problem and mentions the number in the order of ...
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### Finding a function that satisfies the ODE: $-2y'_{(x)}=y_{(\frac{1}{x})}$

I would appreciate if someone could elaborate on how I can find the general function (needs to have a second derivative), that satisfies: $$-2y'_{(x)}=y_{(\frac{1}{x})}$$ I can say that the constant ...
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### Using Laplace transforms, solve for x only in the following pair of simultaneous differential equations [closed]

$$2·dy/dt− y + x·dx/dt - \sin t = 0 \\ 3·dy/dt+ x - 2·dx/dt - e^t = 0$$ initial conditions: given that $x(0) = y(0) = 0$
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### What is it to solve an equation forward?

I'm reading a book in Monetary Economics and I don't understand a step. I have this expression: $$\dfrac{\lambda_{t}}{P_{t}} = \beta \left( \dfrac{\lambda_{t+1} + \mu_{t+1}}{P_{t+1}} \right)$$ And ...
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### Compartmental models

How can we prove the positivity and the boundlessness of compartments in a compartmental model.
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### Help me find error in ODE for sensitivity analysis of parameters of Lotka-Voltera equation

I have a Lotka-Voltera model on which i want to perform parameter estimation by calculating the gradients of the parameters using an extended ODE system. I know there are different methods for doing ...
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### zeros of Bessel function

Let's denote $J_\alpha$ the Bessel functions of first kind, satisfying the equation $$x^2y''+xy'+(x^2-\alpha^2)y=0$$ Now consider its zeros, there are $2$ questions. For the case $\alpha=0$, find the ...
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### Nonhomogeneous First order differential equation

I'm trying to understand what is wrong with this solution, since I'm not getting the same answer in Matlab $y'-xy=xy^{3/2}\, ,y(1)=4$ \begin{align*} y'-xy=&xy^{3/2}&\\ \dfrac{dy}{dx}=&x(y+...
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### A question about change of variables in an ODE or in general

I have a question about differentiation in this question here: differential equation Cauchy-Euler I understand that it uses product rule to go from the 2nd line to the 3rd line (where the arrow point ...
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### Eigenvalue of differential equation [closed]

How can I find the eigenvalues to the linear transformation $$T(y) = y''(t) -2y'(t)$$
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### Find the eigenvalues and eigenfunctions of the following integral equation.

I have the integral equation $$u(x)=1+\lambda \int_0^1 K(x,t)u(t)dt$$ $x \in (0,1)$, $\lambda \in \mathbb{R}$ and $$K(x,t)=\begin{cases} x(t+1) & t \leq x \\ t(x+1) & x \leq t \end{cases}$$ ...
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### Solving $y^{'} + \sqrt{1+{y^{'}}^2}=Ce^{x/k}$

My memories about ODE are rather old. I would appreciate any help driving me to the solution of this equation: $$y^{'} + \sqrt{1+{y^{'}}^2}=Ce^{x/k}$$ where $C$ and $k$ are constants.
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### runge kutta 2 in python

I am trying to solve an equation in fluid mechanics using the runge-kutta 2 method, usually it seems quite doable but in this case its with x y and z and i cant seem to make the code. Here is what i ...
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### Does a restriction to $x$ set when finding $y'$ from solving a second order differential equation affect the domain of the solution? (pursuit curves)

I was solving a problem regarding pursuit curves. The initial differential equation is the following: \begin{equation} xy'' = \sigma \sqrt{1+(y')^2} \end{equation} I used the substitution $v=y'$, thus ...
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### Solve $𝑦''-4𝑦'-5𝑦=1-2x$

Solve $𝑦''-4𝑦'-5𝑦=1-2x$ Please solve the problem in detail (in steps) about the differential equation
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### Evolution equations of a 2-gender age-structure model?

This sounds a bit complicated but I want to grab more feelings on age-structured problems. Less than 2 days to the exam so I appreciate any help. So suppose we get 4 age classes in a population census ...
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### The evolution equations under different year classes of a age-structured model?

This sounds a bit complicated but I want to grab more feelings on age-structured problems. Less than 2 days to the exam so I appreciate any help. So suppose we only get 3 year classes in a school at ...
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### Is my solution correct? I tried solving a homogeneoes system of differential equations through MATLAB.

Given the following equations: These equations represent the movement of a particle. The particle starts moving at $t =0$ in $(-2,0)$ at a speed of $(1,2)$. Solving the problem with the following ...
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### A differential equation of second order

Find $y:I \to \mathbb{R}$ for $I \subseteq \mathbb{R}$, such that $$\begin{cases} y''(t) = \frac{t}{1 + (y(t))^2 + (y'(t))^2} \\ y(0)=1 \\ y'(0)=2 \end{cases}$$ How does one go about solving this? ...
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### Solving coupled ODEs

Context: This is part of one of the derivation I'm trying to do for turbulent wake flows. The book contains the governing equations and the answer but with my limited mathematical knowledge, I'm not ...
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### About Green function on ODE IVP, BVP

I am sophomore student learning ODE. While learning ODE, suddenly met Green function in IVP, BVP. My 1st question is why it is introduced in IVP, BVP, such as: (Due to reduction of order & ...
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### Existence and uniqueness of IVP solutions, vector-valued equations

The existence and uniqueness of a solution to the IVP $$\underline{y'}=f(x,\underline{y}),\enspace \underline y(0)=\underline{y_0},$$ where $\underline{y} = (y_1, y_2,...,y_n)$, is guaranteed if the ...
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### Finding $\lim_{t\to\infty}y$ for the initial value problem $2\frac{dy}{dt}-y=4 \sin(3t)$ with $y(0)=y_0$

I have the initial value problem $$2\frac{dy}{dt}-y=4 \sin(3t), \quad y(0)=y_0$$ I have to determine $\lim\limits_{t \to \infty} y$. I found the solution of the homogeneous problem and then a ...