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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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11 views

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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0answers
17 views

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 ...
2
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2answers
34 views

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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0answers
4 views

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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0answers
10 views

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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2answers
32 views

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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0answers
16 views

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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0answers
9 views

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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1answer
36 views

$y''+a(x)y(x)=0,\quad a(x)$ is continuous function with period $T$.

Consider $y''+a(x)y(x)=0\quad$ where $a(x)$ is continuous function with period $T$. Let $\phi_{1}$ and $\phi_{2}$ be the basis for the solution satisfying $\phi_{1}(0)=1,\phi ' _{1}(0)=0, \phi_{2}(0)=...
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0answers
19 views

Linear ODE $x_t(t,T)-ax(t,T)=-1$, $x(T,T)=0$

I want to find the solution for $x_t(t,T)-ax(t,T)=-1$, where $x(T,T)=0$. That is, for a fixed T, a linear ODE in the t-variable. I don't understand how I could solve this if the lower bound is the one ...
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0answers
29 views

Limits of the wave equation

Consider a wave equation $$\frac{\partial^2 u}{\partial t^2} = c(x)^2 \frac{\partial^2 u}{\partial x^2} \tag{1}$$ We can solve this analytically when $c(x)$ is constant, then the solution is $\sin(x\...
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1answer
32 views

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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0answers
29 views

Find and solve the variational equation for X' = F(X) [on hold]

$F = (x^2 + xy, x + y^3), X = (-1, 1)$ $F(X) = (0,0)$ $A(t) = DF(X) $          $ = $$\begin{pmatrix}2x + y & x\\\ 1 & 3y^2\end{pmatrix}$ $U' = $$...
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2answers
29 views

Obtaining a first-order ODE from a system of ODEs and then proving an asymptote exists

I was debating whether to post this on the mathematics or physics StackExchange, and ultimately, I decided to post this here. I have a system of differential equations which arose from a physics ...
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0answers
44 views

Find the degree of a differential equation

Find the degree of the differential equation. $$(y''')^{\frac{4}{3}}+(y')^{\frac{1}{5}}+ y = 0$$ The answer is available (order = 3; degree = 60). I need help with the steps. I'm stuck in ...
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2answers
54 views

Schrödinger equation involving the Dirac-Delta

I am taking a course on quantum mechanics and I try to understand the time-independent Schrödinger-equation with the Delta-potential: $$\frac{-\hslash^2}{2m}\psi''(x)-V_0\delta(x)\psi(x)=E\psi(x)$$ ...
2
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0answers
43 views

Where do I start on initial value problem $f(t_{0})=x_{0}$ and $f'(t)=h(t)f(t)+g(t)?$

Let $C([a,b])$ be equipped with the essential supremum, and $a, b, t_{0} \in \mathbb R$ where $a < t_{0} < b, h \in C([a,b])$ and $A: C([a,b]) \to C([a,b])$ defined as $(Af)(t):=\int_{t_{0}}^{t} ...
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1answer
17 views

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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0answers
28 views

Transformation of a system of differential equations

Consider the system: $$ \begin{split} \dot W &= i u(t)W(t)\\ \dot S &= e^{-rt}\sqrt{(1-u(t))W(t)} \end{split} $$ I would like to examine a system: $$ \begin{split} \dot x &= u(t)x(t)\\ \...
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0answers
24 views

Derivability of the solution of $\begin{cases}x' = |x| + \lambda e^{\lambda x} \\ x(0) = 1\end{cases}$

Let $\begin{cases}x' = |x| + \lambda e^{\lambda x} \\ x(0) = 1\end{cases}$ it is easy to show that this has a unique solution $X(t,\lambda)$. I need to show that there is $\mu > 0$ for which $X \...
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1answer
25 views

Transforming a nonlinear system to a new system that has an equilibrium point at the origin

Take a look at the following system $$ \begin{align} \dot{x}_1 &= x_2\\ \dot{x}_2 &= -x_1 + x^3_1 - x_2 \end{align} $$ which has three equilibrium points (0,0),(1,0), and (-1,0). In the book ...
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1answer
21 views

Sturm Liouville BVP orthogonality.

ODE is $$ y'' + 3y' + (2+\lambda)y = 0, \quad y(0) = y(L) = 0$$ To show that it is a SLP, I simply did this $$ (e^{3x}y')' + 2e^{3x}y + \lambda e^{3x} = 0 $$ The weight function then is $w(x) = e^{3x}$...
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3answers
45 views

How can I solve $\dot{y} = 2t(y + y^2)$

I am trying to solve $$\dot{y} = 2t(y + y^2)$$ I've seen something like this can be done with a transformation, say: $$ t^\alpha$$ But I can't understand the logic behind the transformation. I would ...
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0answers
24 views

Solving $\begin{cases}x' = t \sin^2(\frac 1 t) - x^2 \\ x(0) = 0 \end{cases}$

Is there a way to compute the solution of the ode: $\begin{cases}x' = t \sin^2(\frac 1 t) - x^2 \\ x(0) = 0 \end{cases}$ where in $t = 0$ we define the field as $-x^2$. I need for an application ...
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2answers
23 views

Differential equations systems

We have that system of differential equations: $$ \left\{ \begin{array}{ll} x'=-x \\ y'=-2y \\ \end{array} \right. $$ I have to solve that system but I only know the method of derive first ...
2
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0answers
47 views

Guiding the solution of ODE with curves

Let $f:\mathbb{R} \times \mathbb{R} \to \mathbb{R}$ be a continuous function such that $\frac{df}{dx}$ is well-defined and continuous on $\mathbb{R} \times \mathbb{R}$. Assume $\gamma \in C^1([0,+\...
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1answer
20 views

How to solve this second order inhomogenous differential equation?

I was wondering how I should go about solving this differential equation. So far I have found that the characteristic equation has a double repeated root of $-1$, so the form will be $Ae^{-t} + Bte^{-...
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1answer
24 views

Why do we find the homogeneous solution of inhomogeneous Differential Equations?

We all know that if we have inhomogeneous differential equation, we must solve for homogeneous solution and the inhomogeneous solution. And, in the end, we add them together for the complete solution. ...
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3answers
45 views

How to solve $\ y'' + y = \sin(x) + \cos(2x) $?

I need to find the solution for $$\ y'' + y = \sin(x) + \cos(2x) $$ general solution is $\ \{ \sin(x), \cos(x) \} $ and trying to "guess private solution: $$\ y_p = Ax \sin(x) + Bx \cos(2x) \\ y'...
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1answer
25 views

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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1answer
34 views

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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0answers
21 views

Inhomogeneous Dynamical System: How to Solve the Integral?

Consider the system of equations $$\begin{pmatrix} x_1 \\ x_2 \end{pmatrix}'=\begin{pmatrix} 1 & 1 \\ 0 & -1 \end{pmatrix}\begin{pmatrix} x_1 \\ x_2 \end{pmatrix}+\begin{pmatrix} e^{-t} \\ -...
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0answers
33 views
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2answers
15 views

I can't figure out how to do undetermined coefficients for low level nonhomogenous diff. equation

I have a very basic problem here that I am unable to solve, and it is not just this problem but many other that require the usage of this method. Consider the following differential equation $y'' -3y'...
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0answers
32 views

explicit solution of the following initial-value problem [on hold]

Find an explicit solution of the following initial-value problem: $$\begin{cases} x^2 y' = y - xy\\ y(−1) = −1 \end{cases}$$
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0answers
14 views

Particular solution inhomogeneous equation

All unknown quantities takes values in $\mathbb{R}^2$ and all equations are defined $\forall t\in\mathbb{R}$. Let $\dot y = By$, where B is the matrix defined as: $$B = \begin{pmatrix} 3 & ...
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1answer
20 views

Green function, determine $D(k)$ in $\,\,\,\,\, -(k^2-m^2)\left[ g^{\mu\nu}- \frac{k^{\mu}k^{\nu}}{k^{2}-m^{2}} \right]D(k)=i\delta^{\mu}_{\rho}$

Given $g^{\mu\nu}=diag(1,-1,-1,-1)$ and $\delta^{\mu}_{\rho}$ the Kronecker delta. I'm in the fourier space: $$-(k^2-m^2)\left[ g^{\mu\nu}- \frac{k^{\mu}k^{\nu}}{k^{2}-m^{2}} \right]D(k)=i\delta^{\mu}...
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1answer
36 views

How to plot phase plane of the ODE system?

I have the system (took very basic example on purpose, to understand the idea): $$\begin{cases} \dot{x} = x \\ \dot{y} = 2x -y \end{cases}$$ so I have plot phase plane. what have been done so far: $...
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1answer
54 views

On the Proof of the Uniqueness/Existence Theorem

I am trying to show that $$|\underline{x}(t)-\underline{y}(t)|\leq \left|\underline{x}(t_0)-\underline{y}(t_0)\right|+\int_{t_0}^{t}\left|\underline{f}(\underline{x},s)-\underline{f}(\underline{y},s)\...
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1answer
38 views

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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1answer
19 views

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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1answer
22 views

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 ...
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0answers
43 views

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 ...
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2answers
33 views

derivation of order 3 method for differential equations

I am stuck with a big problem. Trying to understand the proof that the numerical method of solving differential equation $x_{i+1} = x_i + \tau_iF(t_i+\frac{\tau_i}{2}, x_{i+\frac{1}{2}})$ $...
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0answers
34 views

Another question about solutions to $\frac{f'}{f} = g(x)$.

I have an integration problem which has a singular integrand $\sim \frac{1}{x}$. I would like to make a change of variable so that the resulting integrand is no longer singular. I believe this is ...
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2answers
35 views

Third order Cauchy Euler equation

I need to know how to solve this equation $$x^3y′′′ + 6x^2y′′ + 4xy′ =0$$
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0answers
34 views

Defective differential equation with geometric multiplicity greater than $1$

Suppose I was given a system of differential equation, $$ y'=Ay, $$ where $A\in\mathbb{C}^{n\times n}$ is constant coefficient. Suppose unfortunately, the $\lambda$-algebraic multiplicity $m$ is ...
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0answers
34 views

Phase Diagram for a System of Ordinary Differential Equations

In an economics paper, there is this system of first-order ordinary differential equations: The author then plots its phase diagram for $i(t) = 0$ for $t \leq T$ and reaching $(0,0)$ at $t = T$: My ...
1
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2answers
38 views

Prove $xy^2-e^{-y}-1=0$ is an implicit solution of the differential equation $(xy^2+2xy-1)y'+y^2=0$

Prove $xy^2-e^{-y}-1=0$ is an implicit solution of the differential equation $(xy^2+2xy-1)y'+y^2=0$. I try to implicitly differentiate this equation and it gives $2xyy'+y^2+e^{-y}y'=0$. This doesn't ...
1
vote
0answers
17 views

Truncation error and non self-starting Heun

I've seen two different truncation formulas for the midpoint rule. A common one is $h^3 \frac{ f''}{24}$. Another, referred to as open Newton Cotes, is $h^3 \frac{f''}{3}$. The Newton Cotes ...