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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Is there any theorem for the solutions $u(t)$, $v(t)$ of the following differential equation? [closed]

Suppose I have the following equation (where $a(t)$, $b(t)$ and $c(t)$ are continuous functions) $a(t)(u'(t))^2 + b(t)u'(t)v'(t) + c(t)(v'(t))^2 = 0$ Is there any theorem that could possibly tell me ...
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An inequality for a maximal solution of an IVP [closed]

We have the function $f : \mathbb{R} \times \mathbb{R} \to \mathbb{R}, (x,y) \mapsto \frac{xy}{\sqrt{y^2+1} }$ and the following IVP \begin{align*} y'=f(x,y), \qquad y(0)=1. \end{align*} How does one ...
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Effect of boundary conditions on general solution

I am having problems integrating given boundary conditions on a wave-equation. The problem is as stated below. I am no expert in solving PDE's, so please forgive if I oversee something obvious or &...
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Concept Behind The Equivalence of IVP Solutions to Second Order Linear Differential Equations

A project requires me to establish the equivalence (or lack thereof?) of both solutions to: f(t)=ax''+bx'+cx for x(o) and x'(0). I've realized that I don't actually understand the concept well enough ...
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2 answers
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Find initial value $x_0$ in the PDE: $u_t(x,t) + \partial_x \big(a(x,t)\cdot u(x,t)\big) = 0$ when using methods of characteristic

I am studying PDEs, in particular hyperbolic conservation laws. In particular we are using the method of characteristic to solve some of the problems. Setting Given the quite general PDE $$ U_t + \...
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Solution of IVP is defined for all $t\in\mathbb{R}$ if $f$ is Lipschitz

I'm trying to prove the following statement: Let $f:\mathbb{R}\to\mathbb{R}$ a locally Lipschitz function such that $f(x_1)=f(x_2)=0$ with $x_1<x_2$. If $x_0\in(x_1,x_2)$, then the solution of the ...
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initial value problem and max interval of existence : show that $I^+ \geq 2$

I have the following IVP, $$ \begin{cases} y'(x) &= \left( \frac{y^2(x) + y^4(x)}{1 + y^2(x) + y^4(x)} \right) y^2(x) \\[7pt] y(0) &= \frac{1}{2} \end{cases} $$ on its existence interval $...
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  • 555
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Solving a PDE with a positively valued characteristic coefficient

I want to confirm that I get the solution of the PDE with positive coefficient right. I have the initial value problem: \begin{equation} \begin{cases} u_t=\alpha u_xx \ \ \ \ \ 0<x<L, t>0\\ ...
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1 answer
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If an ODE depends on $f \in C^{\infty}(I, \mathbb{R})$ but not on $f'$, does the same hold for its solution?

Consider an ordinary differential equation (ODE) $$F(t, y, y') =0$$ in the unknown $y \in C^{\infty}(I, \mathbb{R})$. Suppose that $F$ depends on a smooth function $f \in C^{\infty}(I, \mathbb{R})$, ...
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1 answer
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Solving the initial value problem as a vector function r(t) from t

Consider the differential equation $$ \frac{d \textbf{r}}{dt}=(t^3+4t)\,\vec i + t \vec j, \qquad \textbf{r}(0)= \vec i+ \vec j $$ I got $$\textbf{r}(t) = (\frac{t^4}{4}+ 2t^2) \vec i + \frac{t^2}{2} \...
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2 votes
1 answer
108 views

Is the following an Initial Value Problem or not?

I'm trying to solve an Initial Value Problem, but I'm not sure now if the problem I have in hand is even an Initial Value Problem. Notes from Paul Dawkins' Course states IVP has the definition below - ...
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Using D'Alemberts formula for a solution of a general wave equation (without specific I.C.)

So, I have the general wave equation \begin{equation} c^2u_{xx}=u_{tt} \end{equation} with given I.C. : $u(x,0)=g(x)$ and  $u_t(x,0)=h(x)$ I have to use D'Alemberts formmula on the solution. ...
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There is an initial value problem: $x' = (2 \sqrt{|x|} + x^2)(3 - t)$. I need to proof that there is a solution going to infinity in finite time

There is an initial value problem: $x' = (2 \sqrt{|x|} + x^2)(3 - t)$ $x(0) = 0$ Proof there is a solution going to infinity in finite time. Is there an instable, non-negative, global solution? So I ...
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Are BVP and IVP interchangable?

My question is: can the same differential problem (PDE, Action minimization...) be treated as a Boundary Value Problem or as an Initial Value Problem, depending on the nature of the constraints I ...
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2 answers
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Solving a 2nd Order ODE using Finite Difference Method when Mixed Boundary Conditions are given

The problem I'm looking at is $$y'' + 3.05 y' -2.85 = 0 $$ with the boundary conditions $y(0) = 1$ and $y'(1) = 0.0305$. After obtaining the algebraic set of equations using FDM, I'm not sure how the ...
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5 votes
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It is possible to find non trivial solutions $f(t) \in C_c^\infty$ to $\dot{f}(t)=2f(2t+1)-2f(2t-1),\,f(0)=1$ for the whole $\mathbb{R}$ domain?

It is possible to find non trivial solutions $f(t) \in C_c^\infty$ to $\dot{f}(t)=2f(2t+1)-2f(2t-1),\,f(0)=1$ for the whole $\mathbb{R}$ domain? I am trying to find examples of solutions of finite ...
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Properties of solutions of ODE

I am stuck with an exercise. Let $ f = f(t, y) $ be a continuous function such that $ \partial f / \partial y $ is continuous in all points $ (t, y) $. Say what you can about existence of global ...
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  • 528
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Solution to Heat-Like Equation with Diverging Initial Condition

I am trying to solve the equation $$ \frac{\partial}{\partial\alpha}F(x; \alpha) = \lambda \frac{\partial^2}{\partial x^2}F(x; \alpha) \qquad (1)$$ with the condition $F(x; \alpha = 0) = \exp(\ln 2 \, ...
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Exact solution of a BVP of second order

Im solving a BVP which is $y^{\prime\prime}(t)=-y^{\prime2}(t)+y(t)(y^{2}(t)-\frac{3}{2}y(t)+\frac{1}{2})$ with boundary conditions $y(0)=1$ and $y(1)=2$. I need to find the exact solution for this ...
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Solving 1D heat equation with IC and BCs

Suppose I have the heat equation, with IC and BCs: $${\partial T \over{\partial t}}=k{\partial^2 T \over{\partial x^2}}$$ $${\partial T \over{\partial x}}(0,t)=0, \hspace{5mm}T(L,t)=B$$ $$T(0,0)=A, \...
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1 vote
1 answer
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Does $x(t) = \exp\left(\frac{t}{t-1}\right)\cdot\theta(1-t)$ solve $\dot{x}\cdot(1-t)^2+x=0,\,\,x(0)=1$?

Does $x(t) = \exp\left(\frac{t}{t-1}\right)\cdot\theta(1-t)$ solve $\dot{x}\cdot(1-t)^2+x=0,\,\,x(0)=1$ with $\theta(t)$ the standard unitary step/Heaviside function $$\theta(t) := \begin{cases} 0 &...
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1 vote
1 answer
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Solving an initial value problem - PDE

I have to solve $u_{tt}-u_{xx}=0$ with the given I.C.s \begin{cases} u_x(0,t)=u_x(\pi,t)=0\\ u(x,0)=\cos x \\ u_t(x,0)=-\cos x \end{cases} Solving the PDE with separation of variables : \...
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1 vote
1 answer
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How to "formally" prove that $x(t)=\frac{1}{4}(2-t)^2\theta(2-t)$ solves $\dot{x}=-\text{sgn}(x)\sqrt{|x|},\,x(0)=1$?

How to "formally" prove that $x(t)=\frac{1}{4}(2-t)^2\theta(2-t)$ solves $\dot{x}=-\text{sgn}(x)\sqrt{|x|},\,x(0)=1$? (with $\theta(t)$ the standard unitary step function). I have found the ...
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  • 879
0 votes
1 answer
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Show that right interval of maximal IVP solution diverge

I have $f \in C^0(\mathbb{R})$ local lipschitz, $(x_0, y_0) \in \mathbb{R}^2$ and $\lambda_{\text{max}}$ the maximal solution for the IVP, $$ \begin{cases} y'(x) &= f(y(x)) \\ y(x_0) &= ...
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  • 555
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removable discontinuous initial value problem

Suppose I have a initial value problem $$\frac{d}{dt}x(t)=F(x(t)),\;\;x(0)=x_0.$$ and the $F(x)$ is removable discontinuous at finitely many $x\neq x_0$. Also, I know that the problem exists at least ...
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1 vote
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Integral equation corresponding to Initial Value Problem.

The Initial Value problem $$y’’+y=0,y(0)=1,y’(0)=0$$ is equivalent to integral equation $(A)$. $y(x)=1+\int_0^x(t-x)y(t)dt$ $( B)$. $y(x)=1+\int_0^x(t+x)y(t)dt$ $ (C)$. $y(x)=1+\int_0^x(tx)y(t)dt$ $(D)...
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  • 5,278
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0 answers
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Boundary Conditions for a system of PDE

Given a system of following PDEs: $$ u_{x} + v_{y} + 3u-v=0 \\ u_{y} - w_{x}+uw=0 \\ v_{x}-w_{y}=0 $$ I found that the given system of equations is of mixed elliptic-hyperbolic type with the ...
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1 vote
0 answers
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How to find solution to Initial Value Problem of form $x'=Ax+g(t)$

Problem: Find the solution to the following initial value problem $$x'=Ax+g(t), \quad x(0)=\begin{bmatrix} -2\\ 1\\ 4\end{bmatrix},\quad A=\begin{bmatrix} 6 & 3 & -2\\ -4 & -1 & 2\\ 13 ...
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2 answers
35 views

Plot characteristic curves for Initial value problem

I'm trying to plot the characteristic of the following initial value problem, but I am stuck without a curve after finding the characteristic equation. IVP: $$u_t + [u(1 − u)]_x = 0 \text{ for } x ∈ \...
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2 answers
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How to solve homogeneous differential equation with initial value conditions using Green's function?

Solve the differential equation $$xy'' + y' = 0$$ using the Green’s function satisfying the initial condition $y(1) = y'(1)$. Generally, Green's functions are used to solve nonhomogeneous differential ...
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1 answer
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Finite Lenght Wave equation With Only Initial Conditions.

Let $u(x,t)$ be a solution of $$u_{tt}=u_{xx}; 0<x<1, u(x,0)=x(1-x), u_t(x,0)=0$$ Then $u(1/2,1/4)$ is $1$. $3/16$. $2$. $1/4$. $3$. $3/4$. $4$. $1/16$. If i apply D’Alembert formula for Wave ...
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  • 5,278
1 vote
2 answers
176 views

Examples of Finite-Duration solutions to Autonomous Ordinary Differential Equations ODEs?

Examples of Finite-Duration solutions to Autonomous Ordinary Differential Equations ODEs? Examples of the scalar versions: 1st order: $\dot{x} = F(x)$ 2nd order: $\ddot{x} = F(x,\dot{x})$ I have ...
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Uniqueness Cauchy problem autonomous differential equation

I don't know how to solve this problem, if you can help me please: Let be de autonomous differential equation $x'=v(x)$ where $v : \mathfrak{U} \to \mathbb{R}$ is continuous ($\mathfrak{U}$ is an open ...
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Laplace transform on an initial value problem

So i have been trying to solve this Laplace transform for some time now. I have asked my assistant teacher and he also was not able to solve it, so i will try here. problem: $$ y'''- y = -ye^{2t}, \ \ ...
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-1 votes
2 answers
57 views

Approximate the solution of this initial value problem using Euler's method (Maple)

I have the following initial value problem for two functions $y(x)$,$z(x)$: $0=y''+(y'+6y)\cos(z)$, $5z'=x^2+y^2+z^2$, where $0\leq x \leq 2$ and $y(0)=1.7$, $y'(0)=-2.7$, $z(0)=0.5$. Then I got the ...
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0 votes
1 answer
68 views

Solution of system of ODE.

Let $x$ and $y$ be continuously differentiable functions on $[0,\infty)$ satisfying the following differential equations $$\frac{dx}{dt}+(\sin(t)-1)x=\log(1+t),\;x(0)=1$$ $$\frac{dy}{dt}+(\sin(t)-1)y=...
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  • 5,278
0 votes
2 answers
71 views

Characteristics for Burgers equation with $u(x,0)=x$

In the $(x,t)$- plane, the characteristic of the initial value problem $$u_t+uu_x=0$$ with $$u(x,0)=x,0\leq x\leq 1$$ are $1$. parallel straight lines . $2.$ straight lines which intersects at $(0,-1)$...
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2 votes
1 answer
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Why is it true that the solution to this IVP is always $<3$ when $x\geq 0$ (without solving the equation)?

I'm doing some more practice problems for my upcoming DEs test, and I tried this true/false question: The solution to the initial value problem $dy/dx=(x-2)(y-3)^2,y(0)=0$, will always be less than $...
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1 vote
0 answers
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How is Duhamel's Principle a Generalization of Variation of Parameters?

According to Wikipedia, "For linear evolution equations without spatial dependency, such as a harmonic oscillator, Duhamel's principle reduces to the method of variation of parameters technique ...
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  • 1,666
2 votes
1 answer
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Finding characteristics of PDE using method of characteristics

Consider the IVP \begin{equation} xu_x-yu_y=xu\\ u(s,s^2)=1 \; \forall s\in \mathbb{R} \end{equation} I am trying to solve this quasilinear PDE using the method of characteristics, such that I have to ...
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2 votes
1 answer
42 views

Why should I only pick the positive number for $C$ in this IVP (separable differential equation)?

I'm practicing some problems for an upcoming DEs test. I tried the following initial value problem: $\displaystyle\frac{1}{2}\displaystyle\frac{dy}{dx}=\sqrt{y+1}\cos x,y(\pi)=0$ Here's my work: $\...
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1 answer
164 views

Is this initial-value problem separable?

I have $$\frac{dy}{dx} = \frac{2(x^3+x^2-x+1)y}{x^4-1},\; y(0) = 1.$$ I tried separating it and got $2y^{-1}dy = (x^3+x^2-x+1)(x^4-1)^{-1}$ I am unsure if that separation is correct, would anyone be ...
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0 answers
26 views

Solving differential equation of order $1$ using Picars method of consecutive aproximations

Currently I am working with solving differential equations using Picars method. For following example I have to, using Picars method, solve differential equation in 3 iterations: $$y' = x - e^x(x+1)+(...
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  • 503
0 votes
1 answer
51 views

Initial value problem $u_t +b \cdot(Du)+f(x,t)u = h(x,t)$ in $\mathbb{R}^{n} \times (0, \infty)$

Recently, I was able to solve the following (partial differential equations) initial value problem which we will call $(*)$: \begin{cases} u_t+b \cdot (D_{x}u)+cu = h(x,t)&\text{in }\, \mathbb{R}^...
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1 vote
0 answers
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Solve using Laplace and IVP $\displaystyle ay'' + by' +cy = t$ such that $\displaystyle y'(0)=0,y(0)=1$

Let $a,b,c \in \mathbb R$ with $a \neq 0$. Solve the IVP $\displaystyle ay'' + by' + cy = t$ Using Laplace Transform where $\displaystyle y'(0)=0,y(0)=1$ My Attempt $\displaystyle a\mathcal L(y'') + ...
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0 answers
27 views

Number of solutions of initial value problem.

Consider the differential equation $$\frac{dy}{dx}=f(x,y),y(x_0)=y_0$$ I know(Searched on this site If an IVP does not enjoy uniqueness, then it possesses infinitely many solutions) that if $f$ is ...
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  • 5,278
1 vote
1 answer
90 views

Solving separable ODEs

Consider the separable IVP $$\frac{dx}{dt}=f(x)g(t)\;\;\;\text{ with }x(0)=x_0$$ Suppose that functions $f,g,$ and $f'$ are all continuous. We can find the particular solution using the formula $$\int^...
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1 vote
0 answers
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Solution verification of a proof of the Peano existence theorem, using Arzela-Ascoli

$\newcommand{\o}{\mathcal{O}}\newcommand{\d}{\,\mathrm{d}}$I believe what I was required to show is a general version of the Peano existence theorem. Let $\o\subseteq\Bbb R\times\Bbb R^n$ be open and ...
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  • 8,536
0 votes
1 answer
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Do some nonlinear PDE steady state solutions depend on initial conditions (non unique)?

I was told by a colleague that for some nonlinear PDEs the initial conditions can change the steady-state solution. So can a stable steady solution depend on the initial conditions for nonlinear PDEs? ...
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1 vote
1 answer
39 views

Solving a differential equation with initial conditions only on the function

I have the initial value problem $\left\{\begin{gather}E_nf_n(x)+f_n''(x) = 0\\ f_n(-a)=f_n(a)=0 \end{gather}\right.$. Solving it using Laplace transform I get $$f_n(x) = f_0\cos(\sqrt{E_n}x)+\frac{...
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