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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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9 views

Region of convergence of transfer function from differential equation

I learned in my signal processing class that an LTI system can be defined using a linear constant coefficient differential equation. Whenever we have 'initial rest' condition, the LTI system is causal....
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0answers
30 views

ODE with non standard initial condition

I have the following ODE: \begin{equation} f'(x) = \frac{a}{x}f(x) \end{equation} where $a$ is a constant, $x \in [0,1]$. With the following initial condition: \begin{equation} f'(1) = 0 \end{...
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1answer
28 views

Two questions about PDE (sum of solutions)

In an old exercise, I tried to solve a the following problem $$ \begin{cases} u_t + xu_x = u\\ u(0,x)=x^3 \end{cases} $$ I solved the equation with the equalities $$\frac{1}{dt}=\frac{x}{dx}=\frac{...
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1answer
27 views

Numerical Solver Exercise

Sensitivity to initial conditions is well illustrated by a little target practice with your numerical solver. Enter $x' = x^2 - t$ into your numerical solver, and then experiment with initial ...
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1answer
39 views

find the solution of initial value problem

Can you help me with this initial value problem : $\left\lbrace\begin{matrix}(x-t)u_{t}-xu_{x}+u&= &x^{2}+1 \\ u(x,0) &= &1-x^{2} \end{matrix}\right.$ Thanks in advance .
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1answer
50 views

Solving Initial Value Problems [closed]

Considering the following IVP $y’= \frac{1}{1+x^{2}} -2y^{2} \hspace{2cm} x \in (0,1]$ $y(0)=0$ I need to show that the solution is $ \frac{x}{1+x^{2}}$
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1answer
20 views

Lipschitz condition on $x*|y|$

The IVP $y'=x|y|$ is given along with the condition $y(1)=0$. Upon checking the Lipschitz condition one gets, $|x|*||y_{2}|-|y_{1}||\le|x|*|y_{2}-y_{1}|$ Now, if I go locally around $x=1$, $|x|$ ...
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1answer
20 views

Understanding an argument given in my textbook about a second order linear ODE with Dirichlet boundary conditions

Suppose $\frac{d^2y}{dx^2}+\omega y=f(x)$ is given for some well-defined function $f$, with $y(a)=\alpha$ and $y(b)=\beta$. To find the solutions, my textbook splits this into two cases, namely $\...
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0answers
33 views

How do I use the laplace transformations to solve this initial value problem?

So I have been given $\ddot{x} + 8\dot{x} +16x = e^t$ and $x(0) = 0, \dot{x} = 0$. How would I go about solving this initial value problem? As I am unsure of where to begin or what to do. Anything ...
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2answers
45 views

Solving an initial value problem involving the second-order nonlinear ordinary differential equation $y''(x) = y(x) \cdot y'(x) + (y'(x))^2$

I have the following equation $y''(x) = y(x) \cdot y'(x) + (y'(x))^2$ with the initial values $y(1) = 0$ and $y'(1) = -1$. I am seeking some guidance for how to best tackle this particular problem. ...
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1answer
26 views

How do I solve the non-homogeneous wave equation with homogeneous boundary and initial conditions?

I want to solve $$v_{tt}(x,t) - v_{xx}(x,t) = -\left(\frac{3}{4} \cos(t) + \frac{1}{4} \cos(3t)\right) \sin(x)$$ with boundary conditions $$v(0,t) = v(\pi,t) = 0$$ and initial conditions $$v(x,0) = ...
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1answer
66 views

Method of characteristics non-linear PDE

Consider the following initial-value problem: $$xu_x-uu_t=t$$ $$u(1,t)=t$$ I've come to the follow characteristic equations: $$\frac{\mathrm{d}x}{\mathrm{d}\tau }=x,\,\,\,\frac{\mathrm{d}t}{\...
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2answers
32 views

Trying to solve simple pde $u_t = iu_{xx}+2iu$

I'm trying to solve $u_t = iu_{xx}+2iu$ where we know $u(0,x) = \cos(2\pi x)-i\sin(2\pi x)$, $0 \leq x < 1$, $0 \leq t$ with periodic boundary conditions. This is what I tried: Assume $u(t,x) = T(...
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0answers
29 views

PDE-find general solution and solve initial value problem [closed]

I stuck with method of characteristics here, how to find the general solution and IVP? ${u_{xx} + 4u_{xy}+3u_{yy}}={0}, -\infty <x<+\infty, t >0 $ $ u(x,0)=0, -\infty <x<+\infty, $ ...
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1answer
31 views

The Heat Kernel to Solve an Initial Value Problem

The Question I have is: $$u_t(x, t) − ku_{xx}(x, t) = 0$$ $$∀x ∈ \mathbb{R}, t > 0$$ subject to- $u(x, 0) = x^2 − 3x − 1$ $∀x ∈ \mathbb{R}.$ I started off with $$u(x,t)=\int_{\mathbb{R}}e^\...
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0answers
36 views

Using Duhamel's Principle and the Heat Kernel to Solve an IVP

I need to solve this: $u_t(x,t)-ku_{xx}(x,t)=xte^{-t^2}$ By using Duhamel's Principle and the Heat Kernel. So far this is what I've done: $u_t(x,t)-ku_{xx}(x,t)=xte^{-t^2}$ where u(x,0)=0 $u(x,t)...
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1answer
37 views

What is the largest step size h for which the Euler method is stable? (Initial Value Problem)

I have: $$y' = -22*y+3*sin(3*x)$$ $$0 \le x \le 3 $$ $$y(0) = 4$$ as my initial value problem. The question is: What is the largest step size h for which the Euler method is stable, when applied ...
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1answer
58 views

horizontal spring - mass system with damping

Here's the problem I have to solve. I can do the math just fine once I get the IVP set up, but getting it set up is what I don't know how to do. A 2 kg (20 N) mass is attached to a spring, thereby ...
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0answers
44 views

Is that a correct question?

In page no. 132 of the book Linear Partial differential equations for scientists and engineers by Tyn Myint-U, there is following example. Here $|\sin x|$ is not differentiable at $x =n\pi$, $n\in\Bbb ...
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2answers
54 views

Differentiating an “initial value problem”? [duplicate]

I have the following ODE $$y' = 2 - \sin(xy), \qquad 1 \le x \le 3$$ with initial condition $y(1)=-\frac 12$. I have to prove that $|y''(x)| \le 40$ over the domain $[1,3]$. I thought perhaps ...
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2answers
82 views

Prove $|y''(x)| \leq 40$ for all $x \in [1,3]$.

$$y' = 2 - \sin(xy), \qquad\quad 1 \leq x \leq 3, \qquad\quad y(1) = -\frac{1}{2}$$ Attempt:= $$|y''(x)| = |-\cos(xy)(y + xy')| \leq |y + xy'|$$ Not sure what to do next.
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1answer
53 views

Wave Equation with Initial Conditions on Characteristic Curves

I am trying to solve the initial value problem: $$\begin{cases} u_{tt}-u_{xx} =0\\ u|_{t = x^2/2} = x^3, \quad |x| \leq 1 \\ u_{t}|_{t = x^2 / 2} = 2x, \quad |x| \leq 1 \\ \end{cases} $$ I'm unsure ...
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1answer
50 views

One Dimensional Wave Equation with Piecewise Initial Conditions

The problem I am trying to solve is: $$ \begin{cases} u_{tt} - c^{2} u_{xx} = 0 \\ u(x, 0) = g(x) \\ u_{t}(x,0) = h(x) \end{cases} $$ where $h(x) = 0$ and $g(x) = \begin{cases} 0 \ : x < 0 \\ 1 ...
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1answer
23 views

Logistic Population growth, How to find r

The information that I am given: p(12)=95,75 ; P(14)=98 ; and the carrying capacity K = 100. Question : What is r, as defined in: deltaP/P = -r/K * P + r The answer in the book is 0,325 The ...
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1answer
32 views

Initial Value Problem with continuous functions

Hello I haven't seen a question like this before and would appreciate any help or guidance with the question: Let $x, y : I \to \Bbb R^N$ , where $N \in \Bbb N$, be solutions of $$x'(t) = f(t)$$ $$...
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1answer
50 views

Solution of the differential equation $(1+y^2-x^2)y'=\frac{1}{x}, y(1)=1$

Solution of the differential equation $(1+y^2-x^2)y'=\frac{1}{x}, y(1)=1$ is the differential equation have solution ? and is this solution is exist? I am trying to prove by using picard's theorem ...
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1answer
21 views

Trouble Finding a Position Function Given a Dot Product & Initial Value

I tried taking the dot product of r(0) and r'(0) and setting that equal to e^0 = 1 and found that <1,0,0> DOT < a,b,c > = 1 therefore a = 1. But I'm not sure how to proceed from here.
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2answers
47 views

On the smooth dependence on initial conditions for the following ODE

Consider the following ODE: $$ \frac{d^2}{dx^2}f(x)+\frac{3}{2x}f(x)+x^{7/2}\sin f(x) =0\,. $$ This seems to have a non-continuous dependence on its initial conditions. For example, choosing \begin{...
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1answer
74 views

1D Wave equation mixed boundary conditions and I.C.

I have been searching for a solution online, but cannot find one that fits the B.C. and I.C. for this wave equation. I read through this PDF, page 7; although I had similar conditions I just obtained ...
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1answer
29 views

Show that the solution of an IVP eventually leaves a compact set

I am reading a proof which states that a solution $(t,x(t))$ to an IVP ($x' = f(t,x), x(t_0) = x_0$), where $f$ is a function from an open set $U$ to $\mathbb{R^n}$ that is locally Lipschitz ...
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2answers
43 views

Existence of Solution for Initial value Problem $y'(t)=\cos(ty), t>0$ and $y(0)=\frac{1}{n}, n\in\mathbb{N}$ using Ascoli Arzela

Consider the cauchy problem (1)\begin{cases} y'(t)=\frac{1}{1+ty}, & t>0 \\ y(0)=1+\frac{1}{n} & n\in\mathbb{N} \end{cases} (2)\begin{cases} y'(t)=\cos(ty), &...
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1answer
103 views

Solution to 2D- Burger's equation with a source

Does anyone know of any solution to the 2D Burger's equation $$ u_t + (u^2/2)_x + (u^2/2)_y = \beta u $$ $$ u(t=0,x,y) = h_0(x,y)$$ For some constant $\beta $. The closest I've gotten is by following ...
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1answer
38 views

Existence and uniqueness of solution of $y'=(x-y)^{2/3}$ such that $y(5)=5$

Assume the Initial Value Problem: $$ y'(x)=[x-y(x)]^{\frac23}\equiv f(x,y(x)), \quad y(5)=5 $$ Existence: Since $f: \mathbb{R^2} \longrightarrow \mathbb{R}$ and $(x_0,y_0)=(5,5) \in \mathbb{R^2}$, ...
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2answers
42 views

Non bounded ODE solution

Find the value of $w \ge0$ so that the ODE below doesn't have a bounded solution $$y''+y=cos(wt)$$ My attempt: The solution should be in the form $y(t)=Asin(wt)+Bcos(wt)$ $y'(t)=Awcos(wt)-Bwsin(...
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0answers
30 views

Initial value problem related to heat equation.

Let $u(x, t)$ satisfy the IVP: $\frac{\partial u}{\partial t}= \frac{\partial^2 u}{\partial x^2}, x \in \mathbb{R}, t > 0$ and $$u(x, 0) = \begin{cases} 1, \ \ 0\leq x \leq 1\\ 0, \ \ \text{...
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6answers
42 views

ODE with various initial conditions

I encountered the following ODE: $$\frac{dx}{dt} = x(1-x)$$ Of course, this can easily be solved with separation of variables: $$\implies \int \frac{dx}{x(1-x)} = \int dt \implies \ln \bigg|\frac{x}...
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0answers
13 views

Integral representation of time, transforming a graph

I have acquired a solution to a system of differential equations in a parametric form: \begin{equation} x= x_{0}u \end{equation} \begin{equation} y=-x_{0}u+\frac{\gamma}{\beta}\ln u - \frac{C_{1}}{\...
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1answer
15 views

Non-linear IVP with $f(u_1,u_2):=\sqrt{1+\sqrt{u_1^2+u_2^2}}\begin{pmatrix}u_1+u_2\\3 u_2-u_1\end{pmatrix}$

Does the following initial value problem \begin{equation*} \begin{cases} u'(t) = f(u(t)), \\ u(0) = u_0 \end{cases} \quad \text{with} \quad f(u_1,u_2):=\sqrt{1+\sqrt{u_1^2+u_2^2}}\begin{pmatrix}u_1+...
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1answer
39 views

I have $y'(t) = e^{-[g(y)]^2}$

I have $y'(t) = e^{-[g(y)]^2}$, with initial value $y(0) = y_0$, $g: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ and g $\in C^1$. I want to find the limits of $t \to \pm \infty$ taking into account ...
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0answers
12 views

Let $y' = g(t,y)sin^2(y)$, discuting limit of solution.

Let $y' = g(t,y)sin^2(y)$, with $ g: \mathbb{R} \times \mathbb{R}\to \mathbb{R}$, g differentiable. I want to prove that if a solution is maximal, there exists the limits $\lim_{t\to \pm \infty} y(t)$...
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2answers
124 views

Analytical solution of heat equation with non-homogenous boundary conditions

I am trying to get analytical solution of heat equation with non-homogenous boundary conditions, which i can code in MATLAB and compare with my numerical results. In short, i am unable to reach the ...
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0answers
19 views

Extension of the initial value theorem

I'm trying to prove this extension of the initial value theorem for Laplace transforms. If $$\lim_{s\to \infty}s^{v+1}L\{f(t)\}=C$$ , with $L$ the laplace transform then $$D^{v}f(0)=C$$ I know the ...
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1answer
20 views

Uniqueness of solutions of an IVP

I have a misunderstanding regarding a very common reasoning: Let's i.e. look at the IVP $\dot x=f(x), x(t_0)=x_0$ with $f(x)=(x-1)(x-2)$. Now, for $x_0\in ]1,2[$ there can be made an argument that ...
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0answers
61 views

When does any solution of an i.v.p. converges to some steady state?

Preliminaries Let $A = [0, 1]^N$. Consider a dynamical system $\dot{x} = f(x)$, where $x = x(t) : \mathbb{R} \to \mathbb{R}^N$ and $f : A \to \mathbb{R}^N$, where each $f_i$ is in the class $\...
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0answers
18 views

Expression of an initial Gaussian type condition

a friend and I are programming a system of pde's coupled in Mathematica. The initial condition is a Gaussian type perturbation in $r_0$ and width $\sigma$. I think that the initial condition is the ...
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2answers
42 views

Find a set of solutions for $y'=\sqrt{y^2-1},\ y(0)=1$.

For the IVP $y'=\sqrt{y^2-1},\ y(0)=1$ I am supposed to find a set of solutions depending on $2$ parameters. While I can easily find 2 different solutions $y_1(x)=1$ and $y_2(x)=\frac{1}{2}(e^{-x}+e^...
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1answer
27 views

Separating $y$'s and $x$'s

I am doing this initial value problem where I have the equation $y' + \frac{3}{x} y=\frac{\cos(x)}{x^3} $. I know how to do these kinds of problems but I am having trouble getting the $x$ to the right ...
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1answer
28 views

Existence and uniqueness of $\frac{dy}{dx} = \frac{y^2}{x-2}$.

This is from "Introduction to Ordinary Differential Equations" by Shepley Ross. Exercise 1.3.6.b. The problem wants us to apply existence and uniqueness theorem to the initial value problem in below. ...
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1answer
91 views

Describing mappings using dynamics of time-dependent ODE-flows

Let $f\colon\mathbb{R}^n\to\mathbb{R}^n$. When is it possible to find some $g\in C^1([0,1]\times\mathbb{R}^n, \mathbb{R}^n)$, uniformly Lipschitz continuous w.r.t the second argument, such that if $u_{...
2
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2answers
66 views

Well-posedness of heat-equation PDE with only one initial condition

Consider the PDE given by $u_t = \alpha u_{xx}$ with initial condition $u(x, 0) = f(x)$. Now suppose we discretize the problem in the time variable, so we approximate $u_t(x, t)$ by a finite ...