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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19 views

Modelling interest with differential equations (IVP)

Problem : you set a bank account, with initial value k, the bank will pay you continuous interest of 12% per year. a) write an initial value problem for your account balance y(t) after t years Sol: $$...
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16 views

Show this IVP is Lipschitz continuous

I need to show that the IVP $y'=t^3, y(0)=0$ is Lipschitz continuous in $y$. I know I need to find some number $L$ so that $\forall y_1, y_2$ and $t\in[a,b]$, $|f(t,y_2)-f(t,y_1)|\leq L|y_2-y_1|$. I'm ...
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13 views

Proof about existence of two maximal/saturated solutions of an IVP

I have to prove the following: Let $f: \mathbb{R}\times \mathbb{R}\rightarrow \mathbb{R}$ defined by $f(t,x) = x^\frac{2}{3}, \forall (t,x) \in \mathbb{R} \times \mathbb{R}$. Let $x:[-1,0]\rightarrow \...
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6 views

Problem with initial conditions for numerically solving the Sine-Gordon equation using finite differences method (in MatLab).

We are trying to solve numerically a kink solution in matlab using the sine-gordon equation, therefore followed the steps according to the paper (1) listed below. The sine gordon equation is a non-...
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29 views

Series method to solve ODE with initial condition

Using $$y(t) := \sum^{ \infty} _{k = 0} a_kt^k$$ to solve $$y'' - t^2y' = \ln(1+3t)$$ with y(0) = 1 and y'(0) = 0 I first expand $ln(1+3t)$ using Taylor series at t = 0: $$\sum_{k=0}^{\infty}(-1)^{k-1}...
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44 views

Laplace transforms and IVP solutions

When deriving the Laplace transform, we note that the Laplace transform of a function is only defined for $x\ge 0$ (unless in the bilateral case you set $f(x)=0$ for $x \lt 0$). However, we often ...
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1answer
17 views

Maximal Interval of Existence for a particular System of Differential Equations

Consider \begin{align} x' &= (x^2+y^2)\sqrt{|y|}\\ y' &= \frac{1}{y^{42}+1}+\exp(t) \end{align} and let $J \subseteq \mathbb{R}$ denote the maximal interval of existence for the solution ...
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1answer
23 views

Largest interval of validity for the solution of a first-order DE.

I have been trying to solve for the largest interval for which the particular solution of the differential equation \begin{equation*} 9x^2 \frac{dy}{dx} = y^2+3xy-36x^2, \; \; y(-1) = 0 \end{equation*}...
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70 views

How do I know if the ODE has a unique solution?

Given IVP, for $x \in (-5, 5) $ and $ t \in R $ $$ \frac{dx}{dt} = \sqrt{|x|}$$ $$ x(0) = 0 $$ I want to find if it has a unique solution or not. So I use Picard theorem: Either I check if $\sqrt(|...
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61 views

Consider the Initial Value Problem $\frac{dy}{dx} = (y^2-1)\cos(x)$, $y=0$ when $x=0$.

There are no constant solutions to this DE. Can you please explain why the range of the solution of this IVP is a subset of $(-1,1)$. I understand that this is a fairly straightforward IVP to solve, ...
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What is the error of $x_i$ at a fixed time $t_i$ when using Euler's method for the IVP $\frac{dx}{dt}=x$ with $x(0)=1$?

So we have Euler's method: $x_{i+1}=x_i+hf(t_i,x_i)$ where $x'=f(t,x)$ and $h$ is a small step size. Applying this method to $x'=x$ gives $x_{i+1}=x_i+hx_i=(1+h)x_i$. Solving this difference equation ...
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25 views

Coupled nonlinear ODE system

\begin{align*} 2A'+3A^2\bar{A}&=0\\ 2\bar{A}'+3\bar{A}^2A&=0 \end{align*} with ICs $:A(0)=-\frac{\textit{i}}{2}, \bar{A}(0)=\frac{i}{2}$ The solution is given \begin{align*}A(t)=-\frac{i}{\...
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22 views

Separation of Variables Hiccup

Find the temperature $u(x,\ t)$ in a unit length rod modeled by $u_t = 4u_{xx}$ $u(0,\ t) = 0$ $u(1,\ t) = 0$ $u(x,\ 0) = x - x^2$ Breaking out the steady-state temperature, $u(x,\ t) = s(x) + v(x,\ ...
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24 views

Solution to parabolic PDE with drift

Let $\Omega_x:= (-\infty,\infty)$ and $\Omega_t:= [0,\infty)$ be the domains respect the spatial $x$ and time $t$ variable. I'm trying to solve the following partial differential equation for $u:\...
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28 views

Converting integral equation to its primary initial value problem

I converted below initial value problem to Volterra equation of second kind $$ y'(x)-2xy(x)=e^{x^2}, \hspace{3mm} y(0)=1 $$ Supposing $u(x)=y'(x)$ and integrating both sides from $0$ to $x$ yields the ...
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23 views

Finding the domain of definition and the maximum interval of existence of an initial value problem

Consider the following one-dimensional system, dx/dt = F(x), where F : (0,∞) → R is given by, $$F(x) = 1\div2x$$ ∀x > 0. For p > 0, find the solution, u : J → R, of the above ODE, subject to the ...
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10 views

Partial and total derivatives of the solution of an ODE wrt the initial condition

With reference to: Failing to understand some basic idea behind differentiation I thought I understood, but I clearly don't, in fact I have an issue with the following problem: consider the solution $...
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84 views

Properties of the Solutions of $y' = y$

In this section of my notes we take $E(x)$ as a solution of the initial value problem $y'=y , y(0)= 1 $ We show that $E(x+r)$ is also a solution to this problem and show $E(x+r)=E(x)E(r)$. The notes ...
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1answer
29 views

Hyperbolastic rate equation of type II already has its initial condition in it?

I'm modelling some real-world gene expression data with various growth models including linear, exponential, and Verhulst growth but not all of the genes are showing these forms of time-dependence. ...
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18 views

Square shape initial condition in MATLAB

I am analyzing $1$D convection equation in MATLAB using Finite Difference method I would like to start the concentration $C(x,t)$ with a square pulse, and see how this square moves in space and time. ...
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31 views

Non linear spring mass IVP

I’ve read a solution to a question in the Applied Mathematics book , but it seems incorrect to me . So here is the question : Non-dimensionalize and find the leading order solution to the IVP: $ m y’’ ...
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52 views

Solve for x of spring-mass system using method of Laplace transforms

I was told to apply the method of Laplace transforms to solve for x. the displacement $x:=x(t)$ of a spring-mass satisfies the IVP: $$\ddot{x}+4\pi^2x=\sum_{k=1}^3 2\piδ(t-k)$$ for all $t>0$, and $$...
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18 views

Numbers of the solution in a Cauchy problem

I am looking for an example for a real first order ODE which has $0$, $\infty$ or exactly $1$ solution depending on the initial $x(t_{0})$ value. The solution doesn't necessary has to have the domain ...
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1answer
44 views

First-order ODE orbit and time domain

Consider the autonomous Initial-Value Problem (IVP) $$\dfrac {dx}{dt}=\dfrac 12(x^3−x)$$ $$ x(0) =x_0 \in \mathbb{R}$$ where x is a function of time. I am supposed to indicate the time domain over ...
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34 views

Solving IVPs with one-sided initial condition using Laplace Transform

Consider an initial value problem of first order linear ODE. $$\frac{\mathrm{d}}{\mathrm{d}t}y(t)+2y(t)=e^{-t}H(t),\lim_{t \to 0^-}y(t)=2$$ where $H(t)$ is the Heaviside function. Let $L[y: t \to s]=Y(...
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46 views

Existence of successive Picard approximations, $y'=3y +1, \enspace y(0)=2$

I'm trying to show that the successive approximations for $$y'=3y +1, \enspace y(0)=2$$ exist for all real $x$. I think the general formula for approximation $\phi_k$ is $$\phi_k=2+7x+\frac{21}{2}x^2+\...
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16 views

Find the value of $~\lim_{t\to 0^+}u(1,t)~$ where $~u(x,t)~$ be the solution of a given IVP.

Problem: Find the value of $~\lim_{t\to 0^+}u(1,t)~$ where $~u(x,t)~$ be the solution of the IVP $$u_t=u_{xx},~~ x\in \mathbb R, ~t>0\\u(x,0)=\begin{cases} 1 ~~;& 0\le x\le 1 \\ 0 ~...
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7 views

How can change the initial value of my Transfer Function?

A step input of magnitude kss into the transfer function below would generate a simulated response going from 0 to kss value but I want the initial value to be 23(room temperature) so that a step ...
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13 views

PDE's problem for forward traveling wave with initial conditions

$U_{tt}-U_{xx}=0$ $\quad$ for $\quad$ $0<x<\infty$ $\quad$ and $\quad$ $0<t$ $U(0,t)=\frac{t}{1+t}$ $\quad$ for $\quad$ $0 \leq t$ $U(x,0)=U_t(x,0)=0$ $\quad$ for $\quad$ $0 \leq x < \...
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25 views

2nd Order Differential equation with Unit Step and initial conditions [closed]

I have some work but I can't get it to work quite yet, I need to solve this so I can plot it versus time. The function is $90y''+30y'+1000(y-z(t))=0$, $$z(t)= \begin{cases}0 & 0\leq t\leq5/9 \\ -\...
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Existence of $\delta$ for a sequence of initial value problem, where sequence $f_{m}$ converges uniformly on compact subset of the domain.

Let $D$ be an open set in $\mathbb{R} \times \mathbb{R}^{n}$. Let $f_{m}$ be a sequence in $C\left(D ; \mathbb{R}^{n}\right)$ that converges uniformly on compact subset of $D$ to $f_{0} \in C\left(D ; ...
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14 views

Showing Uniqueness of solution for a given Initial Value Problem

I need some help with the following problem Let $t \mapsto\left(x_{1}(t), x_{2}(t)\right)$ be a solution of $$ \left\{\begin{array}{l} \dot{x_{1}}=x_{1}^{2}+\cos \left(t x_{1}\right)-1 \\ \dot{x_{2}}=\...
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39 views

Differential Equation Mixture Problem

A large tank starts with $900L$ of water. At time $t = 0$, a solution with a concentration of $1g/L$ is pumped in at a rate of $10L/min$. The solution is then pumped out at $8L/min$. Let x(t) denote ...
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37 views

The initial value problem $\dfrac{dx}{dt}=x^{\frac{3}{2}}(t),~x(0)=0$ has

The initial value problem $\dfrac{dx}{dt}=x^{\frac{3}{2}}(t),~x(0)=0$ has A) Unique solution B) Two solution C) Infinitely many solution D) None of the above $\textbf{My try}$ As we know for $x=0$ is ...
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1answer
39 views

Does the limit of gradient flow from a given initial value depend on the choice of inner product?

I will start with an example to motivate my question, and then ask it more generally. Example. Let us consider the Hilbert spaces $H^1 \subset L^2$, and a Frechet-differentiable functional $E:H^1\to\...
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1answer
22 views

Two problems on Autonomus ODE properties of unknow solutions

I have to explain a student two exercises but I don't know how continue at the first and how start at the second one. Let $f(x)$ continously differentiable and such that $f(0)=0$. Show that the ...
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20 views

For Method of Undetermined Coefficients, what is the guess for a constant to the power of x? Eg. 2+2^x

I'm doing a initial value problem and need to use the method of undetermined coefficients. I have on the right side 2+2^x, and I'm unsure what guess I would use for this. I've seen every possible ...
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1answer
73 views

advection-diffusion equation with origin-directed drift

Is there a solution (and if yes, what is the solution) to the 3D diffusion equation with a drift velocity directed to the origin $\vec{v}(\vec{x}) = a|\vec{x}|^{-1}\vec{x}$, with $a>0$? Thus a ...
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1answer
59 views

Initial Value Problem to find series solution

Consider the following initial value problem: $$e^{-x}y'' + \ln(1 + x)y' - x^2y = 0$$ $$y(0) = 1$$ $$y'(0) = 2$$ (done) Show that $x = 0$ is an ordinary point in the differential equation. Find the ...
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154 views

Existence and unicity ODE

I've been reading "A text book on ordinary differential equations" of Sahir Ahmad & Ambrossi. On page 37 we have the following excercise: "Explain why $x'+\dfrac{\sin(t)}{e^t+1}x=0$...
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40 views

Differential Algebraic Equations Consistent Initial Condition Guess

Consider following nonlinear DAE with piece-wise nonlinear term. Matrix coefficients are time-invariant and non-negative. Such systems often arise from circuit theory. \begin{gather*} \left[\begin{...
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38 views

What is the period of the solution of a wave equation boundary value problem

In my studies of numerical PDEs, I was given this problem We consider a vibrating string that satisfies the wave equation $u_{tt}=u_{xx}$ on the unit interval with boundary conditions $u(0,t)=0$, $u(...
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19 views

Existence of weak solutions to initial value problems of second-order hyperbolic differential equations on unbounded domains?

Evans gives a theorem for the existence of weak solutions for second-order hyperbolic differential equations. Specifically, he says if $U$ is an open, bounded subset of $\mathbb{R}^n$, we call $U_T = (...
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40 views

Consider the IVP Show that the ODE is exact and compute the solution

Consider the IVP $$ \begin{cases} x'(t) = \frac{-cos(t)x(t)+e^t}{sin(t)},\\ \\ x(1)=0 , \end{cases} $$ Show that the ODE is exact and compute the solution, determining also its largest interval of ...
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1answer
31 views

Compute the solution of the IVP and determine the largest interval

Compute the solution of the IVP $$ \begin{cases} x'(t) = - \frac{t}{x(t)},\\ \\ x(t_0)=x_0, x_0>0, \end{cases} $$ and determine the largest interval where the solution exists and satisfies the IVP,...
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3answers
62 views

Initial value problem $y''=e^{2y}, y(0)=0, y'(0)=1$

Supposedly, it is possible to determine information about the constants of this IVP solution, without computing the solution of the differential equation. Here's how I solve this. Let $z=y'$. In ...
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1answer
26 views

How do initial conditions apply to a general solution for a PDE

Given an equation, $$u_{xx} + u_{xt} -20u_{tt}= 0$$ it can be shown that there exists $\xi = t - 5x$ and a $\eta= t+4x$ such that $$u_{\xi \eta} = 0 \space\space\space\space\space\space\space\space \...
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24 views

How to perform sensitivity analysis to a system of ODEs with respect to its own variables.

The Problem I have a very large system of coupled ODEs which is similar to: $$\begin{multline} \begin{bmatrix}m_1 & 0 \\ 0 & m_2\end{bmatrix} \begin{bmatrix}\ddot{x}_1 \\ \ddot{x}_2\end{...
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1answer
56 views

Can there exist a unique solution to an initial value problem if the hypotheses of the existence and uniqueness theorem are not satisfied?

I have been thinking about this question for a while. I haven't found a definite answer, but I am led to believe that there can be a unique solution to an IVP outside of interval of validity. I just ...
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1answer
33 views

Solving a system of ODEs with a zero eigenvalue and non-zero initial velocity

Consider a system of two second-order linear ODEs for which I have found a solution: $$ \pmatrix{y_1 \\ y_2} = (A_1\cos\omega t + B_1\sin\omega t )\pmatrix{a \\ b} + \pmatrix{C_1 \\ C_2} $$ the ...

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