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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.

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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 ...
Guybrush's user avatar
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2 votes
0 answers
44 views

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 ...
Guybrush's user avatar
  • 327
1 vote
0 answers
54 views

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' \...
ScintillatingWolves's user avatar
2 votes
0 answers
41 views

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 ...
Charles Kim's user avatar
1 vote
3 answers
102 views

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\...
George Lin's user avatar
1 vote
0 answers
26 views

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 ...
user123456's user avatar
1 vote
0 answers
77 views

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 $y \colon \mathbb{R} \rightarrow \mathbb{R}$...
user123456's user avatar
-4 votes
1 answer
56 views

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: ...
somemathperson's user avatar
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0 answers
67 views

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{...
user123456's user avatar
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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 $\...
kerusunox's user avatar
2 votes
1 answer
63 views

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) = ...
Mario Palacios's user avatar
0 votes
1 answer
27 views

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 ...
S M's user avatar
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1 answer
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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 \...
metamathics's user avatar
0 votes
1 answer
28 views

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} \...
gettingmathy's user avatar
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0 answers
21 views

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^...
Christopher Lee's user avatar
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1 answer
39 views

Question about lower bound when solving Variation of Parameter IVPs

Lets say we are solving something like $y'-2xy=2$ for $y(0)=1$ when implementing variation of parameters the standard process is to identify the lower bound in the integral to be 0 so we would be ...
Paul's user avatar
  • 1
2 votes
1 answer
80 views

How to show IVP has unique solution

Given $$ y' = \frac{ty+y}{ty+t}, \ 2\leq t\leq 4, \ y(2) = 4 $$ How do we show the solution and that it is unique. I have been able to get closer to a solution but I have not been able to isolate the $...
xyz04's user avatar
  • 29
3 votes
2 answers
107 views

Discontinuous solution of $y''(x)-2(1-x)(y'(x))^2=0$ with $y(0)=1$ and $y(2)=-1$

I tried to solve the equation $$y''(x)-2(1-x)(y'(x))^2=0$$ with the conditions $$y(0)=1, y(2)=-1.$$ It's easy to verify that the function $$y(x)=\frac{1}{1-x}$$ satisfies both the equation and the ...
Mohamed Mostafa's user avatar
0 votes
1 answer
99 views

Python code for Second Order ODE Initial Value problem using Finite Difference methods

I am trying to solve the following ODE $$\begin{aligned} & y^{\prime \prime}+2 y^{\prime}+y=0 \\ & y(0)=2 \quad y^{\prime}(0)=-1 \\ & 0 \leqslant x \leqslant 1 \end{aligned}$$ Using ...
Yoshiro's user avatar
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0 answers
46 views

How to numerically find initial values of a differential equation that gives us a solution that goes to 0 at infinity.

I am new to numerically solving differential equations and the problem I'm working with involves finding the initial values for a second order differential equation that would give us a solution that ...
Mishi's user avatar
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1 vote
1 answer
48 views

Solving an exact IVP and solution domain

$e^{2y}+x^2+2xe^{2y}\frac{dy}{dx}=0 \\ y(-1)=1$ This IVP is exact and: \begin{align*} \frac{d(e^{2y}x + \frac{x^3}{3})}{dx}=0 \\ \iff e^{2y}x + \frac{x^3}{3}=k; \ k \in \mathbb{R} \\ \underbrace{\...
J P's user avatar
  • 893
0 votes
0 answers
36 views

Finding interval of definition of ODE solutions.

I would like to know how the interval of definition for an initial value problem be determined without explicitly solving it. For example, the solution to $$y' = e^y + 1, \; y(0) = 0$$ is $$ y(t) = \...
Melanka's user avatar
  • 135
1 vote
1 answer
74 views

Equivalence between IVP and integral equation

I was trying to prove the following theorem: If $f(x, y)$ is continuous on some region $R \subseteq \mathbb R^2$ then any solution of IVP $$y'(x)=f(x,y(x)), y(x_0)=y_0 \tag{1}$$ is also a solution of ...
baja1997's user avatar
0 votes
1 answer
56 views

Solution of an IVP through Laplace transform

Let $𝑦(𝑡)$ be the solution of the initial value problem $$y''+4y=\begin{cases} t, & 0\leq t\leq 2\\ 2, & 2<t<\infty \end{cases}.$$ Also, it is given that $$y(0)=0, y'(0)=0.$$ Given ...
PAMG's user avatar
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1 vote
0 answers
56 views

Existence of complex solutions to real IVP

Consider the IVP: $$y'(x)=f(x,y(x))\\y(x_0)=y_0$$ It is known that: If $f$ is continuous and Lipschitz in 2nd variable then: Existence and Uniqueness If $f$ is continuous then: just Existence Now ...
MOMO's user avatar
  • 1,297
1 vote
1 answer
196 views

Solve the initial value problem $y'=e^{-y^2}+1, \, y(0)=0$

Consider the ODE(ordinary differential equation) $$y'=e^{-y^2}+1, y(0)=0.$$ Which of the following statements are true for given ODE? $P: $ The ODE has unique solution on $\mathbb R.$ $Q:$ The ...
neelkanth's user avatar
  • 6,100
0 votes
0 answers
36 views

Show that there's an interval $I \subseteq \mathbb{R}$ with $3 \in I$, so that the initial value problem has a unique solution.

Show that there's an interval $I \subseteq \mathbb{R}$ with $3 \in I$, so that the initial value problem $$ x'(t) = (x(t) - t)e^{t^2+x(t)} \text{ with } x(3) = 7$$ has a unique solution. So far I have ...
Math Wrath's user avatar
0 votes
1 answer
45 views

Initial value problem involving hyperbolic function.

Let $$y(t):[0, \infty)\to \mathbb{R}$$ be the solution of the problem $$y'(t)=(1-y^2(t))\cosh y(t)\ \text{for}\ t>0, \ y(0)=y_0.$$ Find the set of real values of $y_0$ such that $y(t)$ mentioned ...
PAMG's user avatar
  • 4,500
0 votes
0 answers
57 views

Initial value problem with non-local boundary condition

I am solving that problem: $$u_t+u_x+\sigma(x)u=g(x), x\in[0,l], t\ge0$$ $$u(x,0)=\phi(x)$$ $$u(0,t)=\gamma(t)=\int_{0}^{l}\beta(s)u(s,t)ds$$ I have already solved initial value problem with $\gamma(t)...
Kyle Crane's user avatar
4 votes
1 answer
121 views

Proof of Positivity for Solutions in Ordinary Differential Equations (ODEs)

Let $x(t)$ be the solution of the initial value problem: $$ \dot{x}(t) = f(x(t)); \; \; x(0) = x_0 $$ I have made the following asumption during my work: If $x_0 \geq 0$, and $f(0) \geq 0$, then $x(t) ...
Olayo's user avatar
  • 87
1 vote
0 answers
79 views

uniqueness problem with ODE?

Consider the initial value problem $$ \frac{dy}{dt}=4t\sqrt{y}, \quad y(1)=1. $$ This is clearly separable, and using separation of variables we solve $$ \int y^{-1/2} dy=\int 4t dt $$ which gives $2\...
applebees's user avatar
  • 341
3 votes
1 answer
151 views

Why does Wolfram Alpha give a wrong answer to a linear ODE with constant coefficients?

This is very odd as it is a really simple ODE and the interpretation of the input seems correct, yet Wolfram Alpha produces rubbish: $$x'' - 2x' + x = t, \quad x(0) = 1.$$ Wolfram Alpha claims that ...
Klaus's user avatar
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0 votes
0 answers
19 views

What are proper numerical methods to solve IVP at irregular singular point?

I have searched for numerical methods to solve IVP with irregular singular points but didn't find any. What is a proper method I can use to solve such equations? For example, we could consider the ...
Mohamed Mostafa's user avatar
0 votes
1 answer
95 views

Exact solution to system of first-order coupled nonlinear ODEs

I have recently been trying to find an exact solution to the following system of first-order ODEs: $\begin{cases} \frac{dx}{dt}=y(t)*z(t) \\ \frac{dy}{dt}=x(t)*z(t) \\ \frac{dz}{dt}=x(t)*y(t)\end{...
FabrizzioMuzz's user avatar
0 votes
0 answers
50 views

1D heat equation: limit of piecewise continuous function at t=0,x=x_0

$u(t,x)$ in $[0,\infty)\times\mathbb{R}$ such that \begin{cases} \partial_tu-\partial_x^2u=0 \quad in~(x,t)\in(0,\infty)\times\mathbb{R} \\ u(0,x)=\phi(x) \quad for~x\in\mathbb{R} \end{cases} where $\...
tsd's user avatar
  • 21
2 votes
1 answer
67 views

Uniqueness of initial value problem to ODE

I was presented with an i.v.p.:$$y'=\frac{1+y^2}{1+x^2},y(0)=1$$Using separation of variables I obtained $$\arctan(y)=\arctan(x)+c$$ Substituting x=0 and y=1 gives $c=\frac{\pi}{4}$ but if I solve for ...
baja1997's user avatar
0 votes
0 answers
26 views

Unique continuation for first order equation IVP

Let $\alpha \in (0,1)$ and $\eta \in C^{1,\alpha}(-1,1)$ be a solution to $$ \eta'(x) = w(x,\eta(x)),\qquad \eta(x)=0 \quad\mbox{for }x \in (-1,0],\qquad \eta'(x)=0 \quad\mbox{for }x\in (-1,0], $$ ...
Gio712's user avatar
  • 440
2 votes
2 answers
87 views

Mass spring damped system

I have the following problem: A $8\,\mathrm{kg}$ mass stretches a spring $1.96\,\mathrm{m}$. At $t=0$, an external force $f(t)=2\cos(2t)$ is applied to the system. The damping constant is $3\,\mathrm{...
mvfs314's user avatar
  • 2,082
0 votes
0 answers
38 views

Solutions to $y'=y^\alpha$ vary continuously in neighbourhood

Consider the differential equation $y' = y^\alpha$ with initial condition $y(0) = c_0 > 0$. Let $\alpha$ vary in a neighborhood of $1 \in \mathbb{R}$. Of course, for $\alpha = 1$ we get the ...
GreenCoffee248's user avatar
1 vote
0 answers
60 views

Flow is defined using a $C^1$ function is $C^1$

I have a flow defined by the initial value problem: $$\frac{d}{dt}y_t(x)=f_t(y_t(x)), \quad y_0(x)=x$$ where $f_t:\mathbb{R}^k\rightarrow\mathbb{R}^k$. I know the above problem has a unique solution ...
JDoe2's user avatar
  • 766
0 votes
0 answers
33 views

Find a power series solution of the Cauchy-Euler Equation around $x=0$

I have the following problem: Find a power series solution of the Cauchy-Euler Equation around $x=0$: $$(2x+5)y'' + y'= 0, \quad y(0)=2, \quad y'(0)=3.$$ When I tried solving it, the recurrence I got ...
MasterTroppical's user avatar
1 vote
1 answer
114 views

Linear first order PDE: Does a "nice" initial condition guarantee a "nice" solution?

The specific PDE I am interested in is from classical mechanics: consider the phase space distribution function $\rho(\vec q,\vec p,t)$, which satisfies (by Liouville's theorem): $$\partial_t \rho(\...
C.M.O.B.'s user avatar
1 vote
0 answers
48 views

Uniformity of integral curves with respect to initial conditions

Let $U\subset \mathbb{R}^n$ be an open subset and let $X:U\to\mathbb{R}^n$ be a vector field over $U$. Also let $I\subset\mathbb{R}$ be an interval and $\gamma:I\to U$ be an integral curve satisfying ...
Aryan's user avatar
  • 1,528
1 vote
0 answers
72 views

Why, visually, are ODEs not unique without (roughly) Lipschitz coefficients?

$\newcommand{\R}{\mathbb{R}}$ Consider the ODE $$\begin{cases} y'(t) = f(t, y(t)) \quad \\ y(t_0) = y_0 \end{cases}$$ with $f: I \times U \to \R^n$, $t_0 \in I \subset \mathbb{R}$ an ...
Alex's user avatar
  • 637
2 votes
1 answer
108 views

Using Gronwall to prove bi-Lipschitz

I am working through a proof which, for fixed $x \in \mathbb{R}^k$ considers an initial value problem of the form: $$\frac{d}{d t}u_t(x)=v_t(u_t(x)), \quad u_t(0)=x$$ where $u_t:\mathbb{R}^k \...
JDoe2's user avatar
  • 766
3 votes
0 answers
47 views

How can I show that this difference-differential equation has a unique solution?

Consider the following difference-differential equation $$ (1+\alpha t)y_k'(t) - (k\beta+r) y_k(t) = y_{k-1}(t) $$ with $\alpha,\beta,r\in\mathbb R$ and the initial conditions $y_k(0)=0$ for $k\geq1$ ...
Ellenier's user avatar
  • 117
2 votes
0 answers
47 views

proving an ODE with a specific solution can't exist using the existence and uniqueness theorem

I'm having trouble completing the following exercise from my homework: Let $y'' = f(x, y, y')$ be an ODE that fulfills the criteria for the existence and uniqueness theorem (Picard–Lindelöf theorem) ...
OphirW's user avatar
  • 21
0 votes
0 answers
26 views

Issues solving simple separable first order IVP

I have the following IVP: $\frac{da}{dt}=ka^{\frac{1}{2}}$ Subject to $a(0)=a_0$, $t\geq0$, $a\geq 0$. This is a separable equation, which I naively returns the solution: $a(t)=\left(\frac{1}{2}kt + \...
Plagioclase's user avatar
0 votes
0 answers
149 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 ...
Robert Abramovic's user avatar
0 votes
2 answers
95 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 ...
JAEMTO's user avatar
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