# 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 ...
1 vote
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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 ...
1 vote
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### 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 $$c^2u_{xx}=u_{tt}$$ 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 ...
1 vote
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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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### 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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### 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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### 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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1 vote