Questions tagged [initial-value-problems]
This tag is about questions regarding Initial value problems. In the field of differential equations, an initial value problem is an ordinary differential equation together with a specified value, called the initial condition, of the unknown function at a given point in the domain of the solution.
1,135
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Solution Review: Evaluating Monotonicity of Solutions to IVP
I'm studying a problem from an old ODE exam and I have some general ideas on how to solve it but I feel as though these arguments are kind of heuristically clear but wanting in formality. We're given ...
- 327
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44
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Differential Equation Involving Minimum of Two Functions
This one has completely stumped me. It's from an ODE general exam, I'm given the IVP
$$ y' = \text{min}(y^2, M),$$
$$ y(0) = 1.$$
With $M>1$ and I'm asked to give an explicit solution and discuss ...
- 327
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54
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IVP with the Banach fixed point theorem: $y' = \sqrt{x} + \sqrt{|y|}$ and $y(0)=0$
I need to use the Banach fixed point theorem to prove that $y' = \sqrt{x} + \sqrt{|y|}$ (for $x \geq 0$) with the initial condition $y(0)=0$ has a unique solution.
First of all:
\begin{equation}
y' \...
2
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0
answers
41
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Existence of an unique solution of an ODE without boundary conditions
I would like to ask how to determine if there is a unique solution to an ODE that does not have any boundary conditions, nor initial conditions. It sounds weird but I wanted to know if there is a case ...
- 49
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3
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101
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Try to give the solution of PDE with initial boundary
The equation is
\begin{align}
\partial_{t}\!\operatorname{u}\!\left(x,t\right) & =
x^{2}\,\partial_{x}^{2}\operatorname{u}(x,t) +
x\,\partial_{x}\operatorname{u}\left(x,t\right),\quad\quad(t,x)\in\...
- 67
1
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26
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Second order IVP with a parameter
I am working on this problem. I am studying for an exam.
Let $u_{\alpha}$ be the solution of the equation
\begin{equation}
u''(t) + F(t)u'(t) + (u(t))^5 - u(t) = 0,
\end{equation}
with initial ...
- 145
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A second order ordinary differential equation problem
I am trying to solve this problem. I am studying for an exam.
Let $F \colon \mathbb{R} \rightarrow \mathbb{R}$ be a $C^1$ function such that $F(1) = 0$. Let $y \colon \mathbb{R} \rightarrow \mathbb{R}$...
- 145
-4
votes
1
answer
54
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Neural network that learns ODE's 'refuses to learn' initial conditions [closed]
I have implemented a simple network that for now i'm just trying to teach the ODE:
$$\frac{\text{d}x}{\text{d}t} = x$$
Using the simple code below:
...
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0
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67
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Prove existence of solution for IVP
I am trying to solve this ODE problem. I am studying for an exam.
Let $f \colon \mathbb{R}^n \rightarrow \mathbb{R}^n$ be $C^1$ such that $|f(x)| \leq 1 + |x|^{\alpha}$. Consider the IVP
\begin{...
- 145
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40
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continuity of solutions with respect to initial conditions
Let $f(t,x)$ be a continuous function in $a\leq t \leq b, \|{x}\|<H$,
and $x(t)\equiv0$ is the only solution to $$\frac{dx}{dt}=f(t,x),
x(a)=0$$ Prove that given $\epsilon>0$, there exists $\...
- 45
2
votes
1
answer
63
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Find out information about an ODE without actually computing the solution
I'm taking a course on ODEs, and I'm currently on the chapter about initial value problems. In this chapter, we try to find out information about the solutions of the problem
\begin{cases}
x'(t) = ...
0
votes
1
answer
27
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Implementing Initial Conditions in Autonomous Differential Equation
I have the differential equation $ yy'' - 5(y')^2 + y^2 = 0$ with initial conditions $y(0) = 1$ and $\frac{dy}{dx}(0) = 0$ and have been asked to use reduction of order to solve it.
With the ...
- 81
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votes
1
answer
26
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IVP equal to integral equation
I have just recently started getting into differential equations and their solutions. Now I have discovered this theorem:
Let $m \in \mathbb{N}, I=[a,b] \subset \mathbb{R}, f: I \times \mathbb{R}^m \...
- 79
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1
answer
28
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Compatibility of Initial/Boundary Conditions in a Convection-Diffusion Problem?
So, I'm reading a book that numerically solves the following convection-diffusion problem
$$\dfrac{\partial u}{\partial t} + c\dfrac{\partial u}{\partial x} = \alpha\dfrac{\partial^2 u}{\partial x^2} \...
- 91
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How do you find $f'(t,y(t))$ in the context of IVPs?
In the numerical methods class I am taking, IVPs are formulated as $y'=f(t,y(t))$. However, I do not understand how we differentiate $f(t,y(t))$. Here is an example of my confusion:
$f(t,y(t))=y(t)-t^...
- 489