# 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 - ...
15 views

### 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 ...
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 ...
37 views

### 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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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 $F(x; \alpha = 0) = \exp(\ln 2 \, ... 0 votes 0 answers 38 views ### Exact solution of a BVP of second order Im solving a BVP which is$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 ... 3 votes 0 answers 34 views ### Solving 1D heat equation with IC and BCs Suppose I have the heat equation, with IC and BCs: $${\partial T \over{\partial t}}=k{\partial^2 T \over{\partial x^2}}$$ $${\partial T \over{\partial x}}(0,t)=0, \hspace{5mm}T(L,t)=B$$ $$T(0,0)=A, \... 1 vote 1 answer 85 views ### Does x(t) = \exp\left(\frac{t}{t-1}\right)\cdot\theta(1-t) solve \dot{x}\cdot(1-t)^2+x=0,\,\,x(0)=1? Does x(t) = \exp\left(\frac{t}{t-1}\right)\cdot\theta(1-t) solve \dot{x}\cdot(1-t)^2+x=0,\,\,x(0)=1 with \theta(t) the standard unitary step/Heaviside function$$\theta(t) := \begin{cases} 0 &... 1 vote 1 answer 77 views ### 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 : \... 1 vote 1 answer 73 views ### 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 ... 0 votes 1 answer 16 views ### 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, $$\begin{cases} y'(x) &= f(y(x)) \\ y(x_0) &= ... 0 votes 0 answers 21 views ### removable discontinuous initial value problem Suppose I have a initial value problem$$\frac{d}{dt}x(t)=F(x(t)),\;\;x(0)=x_0.$$and the F(x) is removable discontinuous at finitely many x\neq x_0. Also, I know that the problem exists at least ... 1 vote 0 answers 21 views ### Integral equation corresponding to Initial Value Problem. The Initial Value problem$$y’’+y=0,y(0)=1,y’(0)=0$$is equivalent to integral equation (A). y(x)=1+\int_0^x(t-x)y(t)dt ( B). y(x)=1+\int_0^x(t+x)y(t)dt (C). y(x)=1+\int_0^x(tx)y(t)dt (D)... 0 votes 0 answers 28 views ### Boundary Conditions for a system of PDE Given a system of following PDEs:$$ u_{x} + v_{y} + 3u-v=0 \\ u_{y} - w_{x}+uw=0 \\ v_{x}-w_{y}=0 $$I found that the given system of equations is of mixed elliptic-hyperbolic type with the ... 1 vote 0 answers 24 views ### How to find solution to Initial Value Problem of form x'=Ax+g(t) Problem: Find the solution to the following initial value problem$$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 ... 0 votes 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: $$u_t + [u(1 − u)]_x = 0 \text{ for } x ∈ \... 0 votes 2 answers 75 views ### How to solve homogeneous differential equation with initial value conditions using Green's function? Solve the differential equation$$xy'' + y' = 0$$using the Green’s function satisfying the initial condition y(1) = y'(1). Generally, Green's functions are used to solve nonhomogeneous differential ... 0 votes 1 answer 22 views ### Finite Lenght Wave equation With Only Initial Conditions. Let u(x,t) be a solution of$$u_{tt}=u_{xx}; 0<x<1, u(x,0)=x(1-x), u_t(x,0)=0$$Then u(1/2,1/4) is 1. 3/16. 2. 1/4. 3. 3/4. 4. 1/16. If i apply D’Alembert formula for Wave ... 1 vote 2 answers 176 views ### Examples of Finite-Duration solutions to Autonomous Ordinary Differential Equations ODEs? Examples of Finite-Duration solutions to Autonomous Ordinary Differential Equations ODEs? Examples of the scalar versions: 1st order: \dot{x} = F(x) 2nd order: \ddot{x} = F(x,\dot{x}) I have ... 0 votes 0 answers 14 views ### Uniqueness Cauchy problem autonomous differential equation I don't know how to solve this problem, if you can help me please: Let be de autonomous differential equation x'=v(x) where v : \mathfrak{U} \to \mathbb{R} is continuous (\mathfrak{U} is an open ... 0 votes 0 answers 35 views ### Laplace transform on an initial value problem So i have been trying to solve this Laplace transform for some time now. I have asked my assistant teacher and he also was not able to solve it, so i will try here. problem:$$ y'''- y = -ye^{2t}, \ \ ... -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 ... 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 $$\frac{dx}{dt}+(\sin(t)-1)x=\log(1+t),\;x(0)=1$$ $$\frac{dy}{dt}+(\sin(t)-1)y=... 0 votes 2 answers 71 views ### Characteristics for Burgers equation with u(x,0)=x In the (x,t)- plane, the characteristic of the initial value problem$$u_t+uu_x=0$$with$$u(x,0)=x,0\leq x\leq 1$$are 1. parallel straight lines . 2. straight lines which intersects at (0,-1)... 2 votes 1 answer 52 views ### Why is it true that the solution to this IVP is always <3 when x\geq 0 (without solving the equation)? I'm doing some more practice problems for my upcoming DEs test, and I tried this true/false question: The solution to the initial value problem dy/dx=(x-2)(y-3)^2,y(0)=0, will always be less than ... 1 vote 0 answers 35 views ### How is Duhamel's Principle a Generalization of Variation of Parameters? According to Wikipedia, "For linear evolution equations without spatial dependency, such as a harmonic oscillator, Duhamel's principle reduces to the method of variation of parameters technique ... 2 votes 1 answer 44 views ### Finding characteristics of PDE using method of characteristics Consider the IVP \begin{equation} xu_x-yu_y=xu\\ u(s,s^2)=1 \; \forall s\in \mathbb{R} \end{equation} I am trying to solve this quasilinear PDE using the method of characteristics, such that I have to ... 2 votes 1 answer 42 views ### Why should I only pick the positive number for C in this IVP (separable differential equation)? I'm practicing some problems for an upcoming DEs test. I tried the following initial value problem: \displaystyle\frac{1}{2}\displaystyle\frac{dy}{dx}=\sqrt{y+1}\cos x,y(\pi)=0 Here's my work: \... 0 votes 1 answer 164 views ### Is this initial-value problem separable? I have$$\frac{dy}{dx} = \frac{2(x^3+x^2-x+1)y}{x^4-1},\; y(0) = 1.$$I tried separating it and got 2y^{-1}dy = (x^3+x^2-x+1)(x^4-1)^{-1} I am unsure if that separation is correct, would anyone be ... 0 votes 0 answers 26 views ### Solving differential equation of order 1 using Picars method of consecutive aproximations Currently I am working with solving differential equations using Picars method. For following example I have to, using Picars method, solve differential equation in 3 iterations:$$y' = x - e^x(x+1)+(... 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}^... 1 vote 0 answers 78 views ### 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'') + ...
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 ...