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.
1,101
questions
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25
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Checking the Existence of solution to the DE given initial values [closed]
The ODE is as given below:
$$\frac{dy}{dx}=\frac{yx^2-xy^2}{x^3};y(0)=0$$
Following the rule, we take $f(x)=\frac{yx^2-xy^2}{x^3}$
Checking the continuity of $f(x)$, we have;
$$\lim_{(x,y)\to(0,0)}{\...
0
votes
0
answers
79
views
Initial Value Problem for a Particle Subject to a Coulomb Force
Suppose that a particle of charge $q$ is exactly one meter from an oppositely charged particle (same magnitude of charge). The initial velocity of this particle is $0$. From Coloumb’s law and Newton’s ...
0
votes
2
answers
91
views
Solving $x'(t)=\frac{1}{1+x(t)^2} $
I am trying to solve ODE : $x'(t)=\frac{1}{1+x(t)^2} $ with initial value $x(0)=x_0$
I could solve $x'(t)=\frac{1}{x(t)^2}$ by seperating variables.
However, $x'(t)=\frac{1}{1+x(t)^2} $ with initial ...
- 641
-1
votes
0
answers
34
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Initial Value Problem exsist only one solution
The Theorem on the Textbook
Is there anyone who can explain the theorem to me. I find it hard to understand. Thanks!
By the way, what's the formal name of this theorem, I found nothing online.
- 1
0
votes
1
answer
38
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What are the initial conditions for this coupled advection/transport system?
Consider the coupled transport (i.e. advection) system
$$
\begin{align}
\partial_t u + b\partial_x \phi &= 0,\\
\partial_t \phi + b\partial_x u &= 0,
\end{align}
$$
where $u(x,t),\phi(x,t) \in ...
- 38
0
votes
0
answers
24
views
Clarification needed on solving the wave equation with a discontinuous initial condition
I am aware that the wave equation $$u_{xx}=u_{tt}$$ can be analytically solved on a finite spatial domain even if it has a non-smooth initial condition. For instance, one such problem is analytically ...
- 928
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votes
1
answer
55
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Question on the power series method to solve a differential equation
A differential equation $y^{\prime \prime}+\dfrac{y^{\prime}}{z}=0$ with the initial conditions $y(1)=0$ and $y^{\prime}(1)=1$. Power series approach will lead writing $y = f(z)$ as a series centered ...
- 315
0
votes
0
answers
49
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Derivative with respect to parameter of solution of pendulum equation
Consider the pendulum equation $\theta''=a-\sin\theta,\theta(0,a)=\theta'(0,a)=0$. I am trying to find $\frac{d}{da}\theta(t,a)$ at $a=0$.
Taking the derivative of the equation with respect to a gives ...
- 1,841
0
votes
0
answers
37
views
Find all solutions to the Initial Value Problem
We are given that $u$ is the solution to an IVP with the following conditions:
\begin{aligned}
u_{tt} = u_{xx} \\
u_x(0, t) = 0 \\
u(x, 0) = u_0 \\
u_{t}(x, 0) = u_1
\end{aligned}
We are also given ...
- 11
1
vote
1
answer
67
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Initial value problem of $y'(x)=2\cdot\sqrt{|y(x)|}$
I have a problem to solve this initial value problem of $y'(x)=2\cdot\sqrt{|y(x)|}$ with $y(0)=0$.
How to solve initial value problems in general is clear to me (integrate and determine the constant). ...
- 121
0
votes
0
answers
27
views
Bounds on solutions to an ordinary differential equation
For a project I am working on, I am able to show that the derivative of a function $f$ is bounded by:
$$1 - C_1 f(x) \leq f'(x) \leq 1 - C_2 f(x)$$
for some constants $C_1,C_2$. I also know that $f(0) ...
- 1,041
1
vote
2
answers
64
views
Problem solving initial value problems using Fourier transform
Consider the following homogeneous linear system of equations in vector form,
$$
\frac{d}{dt}\mathbf{v}(t) = M\mathbf{v}(t) \hspace{0.6cm} ; \hspace{0.6cm} \mathbf{v}(0)=\mathbf{v}_0
$$
where $M$ is ...
- 278
0
votes
1
answer
48
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Applications of rarefaction waves
I've read quite a few things about rarefaction waves in terms of giving weak and entropy solutions to certain PDE problems with fixed initial time data. I understand that the consideration of these ...
- 185
2
votes
0
answers
36
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Unique solution to a second order IVP
I'm stuck on how to approach this question, any tips/solutions would be amazing.
Consider the initial value problem
$$\frac{d^2\mathbf{x}}{dt^2}=-\frac{GM}{r^3}\mathbf{x}$$
with $\mathbf{x}(0)=\mathbf{...
- 21
1
vote
0
answers
144
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Clarification for "solving the equation for $t < \sigma_0$ is equivalent to solving the reversed-time equation"
I'm trying to understand some theory of linear ODE for control theory and I got stuck.
In the book Mathematical Control Theory, Eduardo Sontag shows that there is a unique solution for the initial ...
- 361
1
vote
0
answers
68
views
Solving IVP using Laplace Transforms
I would like to solve the IVP
$$y'' + 2y' + 3y = 3t$$
with $y(0)=0, y'(0)=1$ using transform methods. Could someone check my work is correct, as my solution differs from that in the text?
Solution.
...
- 5,320
1
vote
2
answers
135
views
If the IVP is x'=1+x^2, x(0)=0, why must the solution be tan(t), (-pi/2<t<pi/2) and not just tan(t)?
I understand that neither $\tan(t)$ nor its derivative $\sec^2(t)$ are defined at $t = k\pi /2$, where $k$ is any integer. If I plug in $x(t) = \tan(t)$, that leaves me with $\sec^2(t) = 1 + \tan^2(t)$...
- 63
1
vote
1
answer
114
views
About solution of equation $y''-C^{2}y=f(x)$
I need some explanation of this result
Problem :
$$\begin{equation}
\begin {cases}
y''(x)-C^{2}y(x)=f(x)~~~~,x\in ]-1,0[\\
y(0)=a\\
y(-1)=b \\
y'(0)=\lambda
\end{cases}
\end{equation}$$
And the ...
- 2,321
0
votes
0
answers
51
views
Examine whether the initial value problem $\frac{dy}{dx}$ = $x\sqrt{y-3}$ , $y(4)=3$ has unique solution or more than one solution or no solution.
My attempt
Let $f(x, y)=x \sqrt{y-3}$ then $f_y (x, y)= \frac{x}{2\sqrt{y-3}}$
We see that $f_y (x, y)$ does not exist at (4,3). We can't conclude whether the solution(s) exists or not.
Solving we get ...
- 21
0
votes
2
answers
56
views
Identifying a differentiable function $\mu$ satisfying $\mu (t)=t^2\int _1^t\frac{1-\mu '(\tau )}{\tau ^3}d\tau $ [closed]
Ascription to an initial value problem.
Identify the differentiable function $\mu :(0,\infty)\rightarrow \mathbb{R} $, which satisfies
$$\mu (t)=t^2\int _1^t\frac{1-\mu '(\tau )}{\tau ^3}d\tau $$
for ...
- 25
3
votes
1
answer
79
views
Showing that an unknown IVP solution is an even function
Given the following IVP:
$$y'=f(x,y)=4x^3 e^{-|x^2+y|}$$
$$y(x_0)=y_0$$
I need to show that the IVP has a single solution in $(-\infty,\infty)$, and that it is an even function.
I was able to show, ...
- 211
4
votes
1
answer
111
views
Existence of unique solution of initial value problem
Let us take initial value problem $y'+\frac{{2}}{{t}}y=4t$, $y(1)=2$ where $p(t)=\frac{{2}}{{t}}$ and $q(t)=4t$. Here $q(t)$ is continuous for all $t$ while $p(t)$ is continuous only for $t<0$ or $...
2
votes
1
answer
44
views
Bounding an unknown IVP solution
Given the Following IVP:
$$y'=f(x,y)=\frac 1 {2+{\cos(xy)}}$$
$$y(0)=\frac 1 2$$
I need to show that the IVP has a single solution $u(x)$ that is defined in $(-\infty,\infty)$, increasing, and that $u(...
- 211
1
vote
1
answer
120
views
Solving a piecewise ODE initial value problem
Find all solutions to the following initial value problem, or show that no solution exists:
$\begin{align}
y^{\prime}=\left\{\begin{array}{ll}
3 x^2-1 & \text { if } x<-1 \\
0 & \text { if }...
- 401
3
votes
0
answers
116
views
Cauchy problem for Partial differential equation.
In my partial differential equation book, there are solved examples of two types of first order quasi-linear PDEs.
First is the Cauchy problem, where the goal is to find the integral surface that ...
- 5,922
0
votes
1
answer
42
views
Trying to solve exercise about Euler method
Help to solve this exercise with Euler Method, my problem here problem is that the solutions I calculated are not close to the real values and they diverge. Thanks!
$$y'=e^y $$
with conditions $$ 0\...
- 13
0
votes
1
answer
77
views
Uniqueness of solution of specific ODE
I'm trying to solve the following problem. It's exercise 1.6 of the book Ordinary Differential Equations by Barreira & Valls.
Let $f:(a,b) \rightarrow \mathbb{R} \backslash \{0\}$ be a continuous ...
- 35
1
vote
1
answer
62
views
Statement about inequalities of two distinct solutions of an ODE
I have the following initial values problem:
\begin{equation}
\left\{\begin{array}{@{}l@{}}
y'(t)= t\tan(y(t) - \frac{\pi}{4})\\
y(0) = \frac{\pi}{2}
\end{array}\right.\,.
\end{equation}
...
- 11
0
votes
3
answers
84
views
Solve initial value problem $\frac{dy}{dx}=x^2(1+y), y(0)=3$
We have
$\int\frac{1}{1+y}dy=\int{x^2dx}$,
$\ln|1+y|=\frac{x^3}3+C$
$C=\ln4$
The solution from my textbook is $y=4e^{\frac{x^3}3}-1$, which implies that $1+y$ is always positive. Why is it positive?
- 726
1
vote
0
answers
65
views
Why infinitely many solutions for $a=0$? [duplicate]
The question states
The Initial value problem $y'=2\sqrt{y},y(0)=a$ has
(a) unique solution if $a<0$
(b) no solutions if $a<0$
(c)infintely many solutions for $a=0$
(d)unique solution if $a\geq ...
- 1,125
1
vote
0
answers
33
views
How to solve this initial value problem? Or is it really an initial value probem?
I have peculiar kind of initial value problem, and I don't know how to proceed. The problem is as follow:
$$\frac{dw(\epsilon,t)}{d\epsilon}\bigg|_{\epsilon=0}=f(t).$$
Here $w(\epsilon, t)$ is ...
- 734
1
vote
1
answer
45
views
Definition of stability for multistep methods
I am reading An Introduction to Numerical Analysis by Atkinson, and I want to make sure that I am correctly interpreting the author's definition of stability for a multistep method. In this context, a ...
- 1,087
1
vote
0
answers
27
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Reference request for the analysis of a nonlinear Fokker-Planck type PDE
It is well-known that the linear Fokker-Planck equation (written in one space dimension for simplicity) $$\partial_t \rho = \partial_x \left(\rho_\infty \partial_x\left(\frac{\rho}{\rho_\infty}\right)\...
- 2,724
1
vote
1
answer
92
views
Global solution of IVP.
My question is as follows :
Let $f$ and $g$ are positive continuous functions on $\mathbb R$ with $g\leq f$ everywhere. Assume that IVP $$x’=f(x), x(0)=0$$ has a solution defined on all of $\mathbb R$...
- 5,922
3
votes
2
answers
177
views
Derivation of an asymptotic error formula for the Trapezoidal method for IVPs
I am trying to prove the following theorem, which is a paraphrased version of Exercise 6.18 in Kendall Atkinson's An Introduction to Numerical Analysis:
Theorem. Let $[x_0,b]$ be a finite interval, ...
- 1,087
1
vote
0
answers
29
views
Stability of the Trapezoidal method for solving IVPs (Proof verification)
In Kendall Atkinson's An Introduction to Numerical Analysis, the author proves that the explicit Euler method for solving an inhomogeneous IVP is stable with respect to perturbations to the vector ...
- 1,087
3
votes
0
answers
72
views
Showing that if $f \in C^k$ and $\dot{x}(t) = f(x(t),t)$, then $x \in C^{k+1}$ (Proof verification)
I came across the following easy fact (Lemma 3.17) in Meiss' Differential Dynamical Systems. Here is my paraphrased version:
Lemma. Let $E \subseteq \mathbb{R}^d$, let $I \subset \mathbb{R}$ be a ...
- 1,087
0
votes
1
answer
74
views
Proof that the Trapezoidal Method for solving IVP is $O(h^3)$
My question comes from Chapter 6 of the book Introduction to Numerical Analysis, 2nd edition, by Kendall Atkinson. In section 6.5, p.368-369, the author proves that the Trapezoidal method for solving ...
- 1,087
1
vote
0
answers
54
views
Characterizing the behavior of the solutions of $y’’-q(x)y=0, y(0)=a \ne 0, y'(0)=1$
Consider IVP $$y’’-q(x)y=0, y(0)=a, y’(0)=1$$ where $q(x)$ is continuous and positive on $\mathbb R$. Then my observation is as
If $a>0$, every solution is convex and hence strictly increasing on $[...
- 5,922
2
votes
1
answer
104
views
On number of solutions of ODE $y’=-y^a, y(0)=0$.
How many solutions the ODE of the type $y’=-y^a, y(0)=0, 0<a<1$ have? I am able to prove that the IVP $y’=y^a, y(0)=0$, where $0<a<1$, has infinite
number of solutions by finding it’s ...
- 5,922
0
votes
0
answers
39
views
ODE with tridiagonal anti-Hermitian matrix and initial condition in one entry
I am given an anti-Hermitian and tridiagonal matrix $\bf M$ and I need to solve for a vector $\bf v$ such that
$$ \left[ \exp( t {\bf M} ) \, {\bf v} \right]_{1} = {\bf 0} $$
or at least $\approx {\bf ...
- 33
0
votes
0
answers
42
views
How to properly lift an ODE from $\mathbb{R}^n$ to $\mathbb{R}^{n+1}$?
The following ODE consists of a sum of two time-dependent vector fields $u$ and $v$:
\begin{align}
\begin{cases}
d\mathbf{x}_{t} & = \left[ u(\mathbf{x}_{t},t) +v(\mathbf{x}_{t},t) \right] dt \\...
- 123
0
votes
0
answers
39
views
PDE with initial condition via method of characteristics
Solve $xu_x+(x+2y-2)u_y=x+y, \space x>0, \quad u(1,y)=y$
My attempt:
$x'(t)=x(t) \space \Rightarrow \space x(t)=c_1 e^t$
$y'(t)=x(t)+2y(t)-2 \space \Rightarrow \space y'(t)-2y(t)=c_1 e^t-2 \space \...
- 13
1
vote
0
answers
63
views
Taylor expansion of PDE
I have the following PDE
$$\frac{\partial u}{\partial t}(r,t)=\frac{1}{r^{2}}\frac{\partial}{\partial r}\left(r^{2}\frac{\partial u}{\partial r}(r,t)\right),$$
for which I have determined a numerical ...
- 407
1
vote
0
answers
52
views
A system of PDEs where the solution of second equation is a parameter of the first equation
Let $\Omega \subset \mathbb{R}^n$ and $T>0$. Consider the following system of two equations where the unknowns are $u$ and $v$
\begin{equation}
a(x,t) \partial_t u + \Delta (v \Delta u) - \nabla \...
- 75
0
votes
0
answers
37
views
Nonlocal linear approximation to nonlinear ordinary different equation
Suppose I have a nonlinear ordinary differential equation, in several variables, with a stated initial condition. How would I go about finding a nonlocal linear approximation? What is known about such ...
- 238
0
votes
1
answer
30
views
Determining the change of Error when changing the step size $h$ of the Taylor Method for IVPs
I am given the following IVP that I am to solve numerically using the Taylor method of order 2:
\begin{equation}
\left\{ \begin{array}{cc}
y' = \underbrace{ye^x + 1}_{f(x,y(x))}, & \; 0\leq x \...
- 103
0
votes
2
answers
132
views
IVP $\frac{dy}{dx}=-\sqrt{y}, y(0)=0.$
How many solutions does the differential equation $\frac{dy}{dx}=-\sqrt{y}, x>0$ with the initial condition $y(0)=0$ have? It is clear that zero is one of its solutions. However, when we attempt to ...
- 5,922
1
vote
1
answer
119
views
Solution of Differential equation $ 4x^2y''+8xy'+y=2xy^3 $
Find solution of differential equation
$$ 4x^2y''+8xy'+y=2xy^3, \quad y(1)=1, \ y'(1)=0 $$
I see that $ 4x^2y''+8xy' = (4x^2y')' $, but don't know, if it is useful or not
If we divide by $ y^3 $ ($y \...
- 31
0
votes
0
answers
30
views
Are boundary value problems with homogeneous time dependent boundary conditions never separable?
Consider the following initial-boundary value problem for the heat equation:
$$u_t(x,t)=u_{xx}(x,t),\ \ \ x\in[0,1] \\
u(x,0)=u_0\\
a(t)u(0,t)+b(t)u'(0,t)=c(t)u(1,t)+d(t)u'(1,t)=0$$
Meaning, our ...
- 4,158