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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15 views

How can i solve an initial value problem with Euler's improved method? [closed]

I was given an initial value problem : $$x^2y''-2xy'+2y =x^3lnx\ \ \ \ \ 1<=x<1.2 \\y(1)=1 , y'(1)=0 \ \ \ \ h=0.1 $$ And i want to solve it with Euler's impoved method . Any help would be ...
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1answer
20 views

Solve Wave Equation Initial-Boundary-Value-Problem

I am trying to solve the following problem and this is my working so far. I'm struggling to get to the general solution for $X(x)$ as I'm not sure of the $\lambda$ value. Please could someone point ...
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1answer
20 views

What is the largest open interval $I$ containing $x = 0$ on which $\exists !$solution $y(x)$ to this problem? (Understanding the solution)

Consider the initial value problem: $(\sin(x)-1)y''' + (x^2-x)y'' + 1\frac{1}{(x-1)}y' + x^5y = e^{x^3}$ $y(0) = 1, y'(0) = 5, y''(0) = 2$ What is the largest open interval $I$ containing $x = 0$ ...
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56 views

Initial-value problem by separation of variables

Sorry if this is too easy, but I have really been struggling with this. I have been asked to solve the initial-value problem by separation of variables and determine the maximum time interval of ...
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14 views

Formulae for Input-Free Solution

I have the formula $y_i = y_\delta' y(0) + y_\delta y'(0) + a y_\delta y(0)$ to find the input-free solution to an Initial Value Problem $y'' + ay' + by = f$, where $y_\delta$ is the solution to $y_\...
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2answers
23 views

Solve $u_t+\sin(t) u_x=0$ with initial condition $u(x_0,0)=x_0^2$

I'm starting to learn PDEs and I'm trying to solve the transport equation $u_t+\sin(t) u_x=0$ with initial condition $u(x_0,0)=x_0^2$ by the method of characterisitics. So I define $\xi'(t)=\sin(t)$ ...
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Peano's Existence Theorem. How to find the interval I on which the solution is defined

Peano's theorem states (more or less) that an IVP in which x'=f(x,t) with f continuous on its open domain has at least one solution $\phi$ defined on I=[$t_0$-h,$t_0$+h]. However my book doesn't ...
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1answer
23 views

Solution of the quasilinear equation $p - z q + z = 0$ with initial data curve

Hello ı cant write like maths notation hear sorry for that. Find the solution of the initial value problem for the quasilinear equation $p - z q + z = 0$ for the initial data curve $\Gamma: x_{0} = 0,...
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Initial Value Problems d'Alembert's formula

$$u_{tt}-u_{xx}=0, -\infty<x<\infty,t\ge0$$ $$u(x,0)=1, u_t(x,0)=x(x-4)$$ My attempt, solving by d'Alembert's formula: $u(x,t)=\frac{1}{2}[(f(x+ct)+f(x-ct)]+\frac{1}{2c}\int_{x-ct}^{x+ct}g(x)...
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1answer
12 views

Initial-Boundary Value Problems

$$u_{tt}-4u_{xx}=sin(3x\pi)-7sin(5x\pi),0\le x\le1,t\ge0$$ $u(0,t)=0$, $u(1,t)=0$ $u(x,0)=0$, $u_t(x,0)=0$ I'm having a hard time solving this question. My first attempt was writing $u(x,t)$ as $\...
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1answer
27 views

Consider the initial value problem $y'=y+x, y(0)=2$. Find the first $4$ terms of the Taylor series.

I'm not sure I'm understanding the question quite right. What I did is: \begin{gather} y'=x+y, \quad\quad y'(0) = 0 + y(0) = 2\\ y''=1+y',\quad\quad y''(0) = 1+y'(0) = 3\\ y'''=y'',\quad\quad y'''...
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1answer
21 views

Using Picards theorem to show that the initial value problem has a unique solution

I am trying to show that the IVP $$x'=\sqrt{x(t)}+1, t\in[0,1],\\x(0)=0, (t_0=0)$$ has a unique solution and show whether the initial value problem satisfies the assumptions of Picard’s Theorem, ...
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1answer
16 views

Cauchy Functions And Initial Conditions

I'm having a hard time solving this problem. $$x''+4x=t$$ $a)$Derivative the corresponding Cauchy fn. $b)$Find the solution of the given equation to the I.C., $x(0)=0$, $x'(0)=0$ So the char. eq. ...
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1answer
63 views

Wave equation with Neumann boundary condition

I have the following problem: $$ \begin{array}{ll} &u_{tt}(x,t)=4u_{xx}(x,t),&x>0, t>0\\ &u_x(0,t)=-\cos(t),&t>0\\ &u(x,0)=e^{-x},&x>0\\ &u_t(x,0)=2e^{-x},&...
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15 views

Boundary condition for a well-posed linear, inhomogeneous, second-order PDE

Unfortunately, my exposure to partial differential equations in mathematical physics has been very limited and hence this question. Consider a general linear, inhomogeneous, second-order, partial ...
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23 views

Can this relation be expressed in closed-form?

I am trying to simulate some process with parameters $t_i$, $T_i$ and $U_i$ that depend on some counting variable $i \in [0,n]$ where $n$ is the number of intervals I take for my simulation. The ...
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1answer
31 views

What is the approximation obtained with the midpoint method after one step, answer is a function of h?

Write down the approximation obtained with the midpoint Runge Kutta method after one step (the answer will be a function of h). I am stuck on this part for a first-order system. I'm not sure how to ...
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0answers
62 views

Finding the interval of definition for a IVP

Given the differential equation: $$x^3y''−xy'+y=\frac{1}{x^2}e^{−\frac{1}{x}}, y(1)=y'(1)=\alpha$$ And the general solution of the differential equation to be: $$y(x)=c_1x+c_2xe^{\frac{−1}{x}}−\frac{e^...
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15 views

Existance and uniqueness of solution in a proper neighborhood

Let $x_0 \neq 0$. Prove that the following Cauchy problem $\begin{cases} x'= \frac{t+x^2}{t-x}\\ x(0) =x_{0} \end{cases}$ has a unique solution in a neighborhood of $t=0$ and that this solution is ...
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3answers
38 views

Solving $(\sqrt{xy}-x)\,dy+y\, dx=0$

I am trying to solve the initial value problem $$ (\sqrt{xy}-x)\,dy+y\, dx=0 $$ $$ y(1)=0 $$ I have done the following: $y=xu$, $dy= u\,dx+x\, du$, $$ (\sqrt{x^2u}-x)(u\,dx+x\, du)+xu\, dx=0 $$ $$ (|x|...
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0answers
46 views

IVP with boundary condition (incorrect statement)

Suppose, I would like to solve initial value problem (IVP) $$ (1):\quad \begin{cases} x'(t)=f_1(t,x,y),\\ y'(t)=f_2(t,x,y), \end{cases} $$ with initial conditions $$ (2):\quad x(0)=x_0,y(0)=y_0 $$ ...
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2answers
40 views

Transform second order IVP to first order

IVP is given by: $$y''(t)= \dfrac{1}{1+t} \\ y(0)= y'(0)=0 $$ Need to transform this IVP into $$x'(t)=Ax(t)+b(t) , x(0)=0 \tag 1 $$ I am having some issues with this specific question. I am able ...
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1answer
37 views

Proof of Existence and Uniqueness of IVP (Exam Question)

I am trying to practise some past exam questions but there are no solutions online. Here attached is the following question. This is an undergraduate course. Could anyone help me how to solve this? I ...
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13 views

Use Green's function for second order ODE

Suppose we have a second-order ordinary differential equation with initial conditions, i.e., $u'' + a u' + b u = f$ with $ u(0) = c$ and $u'(0)=d$ I know this can be solved by splitting the problem ...
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0answers
32 views

IVP defined on a maximal interval

I am sticking with the following problem: Suppose $U=\mathop{R} \times \mathop{R}^n$ and that $$|f(t,x)| \leq g(|x|)$$ for some positive continuous function $g\in C([0,\infty)]$ which satisfies $$\...
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1answer
37 views

Why is it false that for all $y\in\mathbb{R^n}$ the solution of the initial value problem $x(0) = y$ exists for all time $t$.

Is the following statement false because our solutions for the initial value problem may not exist when $t=0$; depending on our function? Also, uniqueness does not exist if the system is nonlinear, ...
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2answers
69 views

Initial value problem $y' = y(2-y)$

I wish to solve the differential equation, $$\frac{dy}{dx} = y(2-y)$$ with initial condition $y(0) = 1$. I'm new to differential equations, and I've never seen an equation with only one variable like ...
1
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1answer
25 views

IVP using system of equations: Undetermined Coefficients

The given equations and values are $$x^{"}=-5x+2y\space(1)$$ $$y^{"}=2x-8y \space(2)$$ $$x(0)=10 \space,x^{'}(0)=13$$ $$y(0)=5 \space ,y^{'}(0)=-16$$ Rewriting $(1)$ and $(2)$ $$(D^2+5)x-2y=0 \space (...
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0answers
20 views

Heat equation with inhomogenious boundary conditions

I'm trying to find the analytic solution for this problem: $\frac{∂u}{∂t} =\frac{∂^2u}{∂x^2}, 0\leq x\leq 1,t\geq 0,$ $u|_{t=0}=x+sin(3\pi x), 0\leq x\leq 1,$ $\frac{∂u}{∂x}|_{x=0}=1, t\geq 0,$ $...
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1answer
29 views

Solve the 2nd order ODE IVP using method undetermined coefficients

The given equation and values are $$y^{"}-2y^{'}+y=10e^x$$ $$y(0)=4 \space , y^{'}(0)=2$$ Finding the root's is easy enough: $$(m^2-2m+1)=0$$ $$m_1=1$$ Then our Aux. Equ. is: $$y_c=C_1e^x+C_2xe^x$$ ...
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1answer
14 views

Picard-Lindelof Iteration to solve IVP

We have the IVP $$\dot{y}(t) = Ay(t)$$ $$y(0)=c$$ We consider the Picard-Lindelof Iteration. One step of this iteration is given by $$y_{k+1} = y_0(t) + \int_{0}^{t} A y_k (\tau) d\tau$$ where $y_0(...
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1answer
37 views

Solving the following Initial Value Problem

Solve the initial value problem: $$y_1'=2y_1+2e^{2t}$$ $$y_2'=3y_1+2y_2+3e^{2t}$$ and $y_1(0)=2, y_2(0)=3$ I'm trying to start by using $y'=Ay+g$ Does $g=\begin{bmatrix}2e^{2t}\\3e^{2t}\end{bmatrix}$...
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1answer
20 views

Explicit and Implicit Euler-Method for $\dot{y}(t) = - \lambda y(t), y(t_0) =y_0$ and $\dot{y}(t) = - t (y(t))^2, y(t_0) =y_0 > 0$

We consider the two IVP $$\dot{y}(t) = - \lambda y(t), y(t_0) =y_0$$ $$\dot{y}(t) = - t (y(t))^2, y(t_0) =y_0 > 0$$ We're asked to execute one step with step-size $h$ with the Explicit and ...
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1answer
26 views

Prove that no solution exists for $ \frac{dx}{dt} = f(x), x(0)=0,$ where $f(x) =1$ if $x<0$ and $= -1 $ if $x \geq 0$.

Consider the IVP given by $ \frac{dx}{dt} = f(x), x(0)=0,$ where $f(x) =1$ if $x<0$ and $= -1 $ if $x \geq 0$ There exists no solution to this IVP on any interval $[0, T], T> 0$. Incorrect ...
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1answer
61 views

Series solution of IVP $y' = \sqrt{1-y^{2}}$

Given the IVP $$ y' = \sqrt{1-y^2},\qquad y(0) = 0 \tag{1}$$ I'm looking for a solution in the form of a power series in power of $x$ about $x=0$. Particularly, I'm looking for the coefficients up to ...
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1answer
36 views

Error upper bound using Euler's Method

Determine an upper bound on the error made using Euler's method with step size $h$ to find an approximate value of the solution to the initial-value problem: $\frac{dy}{dt} = t - y^4$, $y(0) = 0$ at ...
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1answer
20 views

Laplace transform of Legendre's equation, differential form

I'm trying to find a differential equation involving $Y(s) = \mathcal{L}[y(t)]$ of the Legendre's equation $$ (1-t^{2})y'' -2ty' + \alpha(\alpha+1)y = 0\qquad\qquad ,y(0)=1, \quad y'(0)=1,$$ for some ...
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1answer
37 views

Bound of an ODE using the Comparison Theorem

Given the IVP: $\space y'(x)=y^2-x , y(0)=1$ I need to prove using the comparison theorem that for all $x\in[0,1): 1+x\le y(x)\le \frac{1}{1-x}$ The comparison theorem I'm referring to is this one:...
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1answer
28 views

Convert $\dddot{y}(t) = (\dot{y}(t)- y(t))^2 + 3\sin{(t)}y(t)$ and $\ddot{y}(t) = \dot{y}(t) -y(t)^2$ into 1. Order IVP.

1) We're given the IVP 3. Order $$\dddot{y}(t) = (\dot{y}(t)- y(t))^2 + 3\sin{(t)}y(t)$$ with initial values $y(0)=a, \dot{y}(0)=b, \ddot{y}(0)=c$ and want to convert it into a 1. Order IVP. 2) We're ...
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1answer
44 views

NUMERICAL ANALYSIS: HEUN'S METHOD

so I did the loglog plot of the maximum error vs the time step when using Heun's method. I see that for very small step sizes the order gets disrupted. Is there a specific reason for that? I also did ...
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1answer
27 views

How to show that the initial value problem has a unique solution in the given interval?

Use Picard’s theorem to show that the initial value problem $(1+e^x)\frac{dy}{dx} = \sin(x + y^3)$, $y(1) = 3$, has a unique solution on the interval $x ≥ 1$. By Picard's Existence and Uniqueness ...
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1answer
36 views

Why are maximal existence intervals of IVP open?

In my current ODE lecture we recently introduced the proposition Let $D \subseteq \mathbb{R} \times \mathbb{R}^{n+1}$ be open, $f: D \rightarrow \mathbb{R}^n$ be continuous and admit a local ...
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0answers
23 views

Solving a complicated system of ODEs in R, unsure if code/ method is correct

Apologies upfront for such a large question. Attached is a write up of the problem in case any further info is needed. These are the equations I am trying to solve: $$\frac{d}{dx}U_n(x)=1-U_n(x)\frac{...
2
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1answer
32 views

Solving IVP with Laplace transform involving step function and summation

Given the IVP $$ y'' + y = f(t) , \qquad\quad y(0) = 0 , \quad y'(0) = 0 , \tag{1}$$ where $$ f_{k} (t) = u_{0} + 2 \sum_{k=1}^{n} (-1)^{k} u_{k \pi}(t). \tag{2}$$ We want to find the solution. My ...
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1answer
18 views

3rd order ODE: System of Linear Eq. from a initial value problem

The given equation and initial values are $$y^{'''}+12y^{''}+36y^{'}=0$$ $$y(0)=0$$ $$y'(0)=1$$ $$y''(0)=-7$$ Using the auxiliary equation and factor we get $$m_1=0, \space m_{2,3}=-6$$ Then the ...
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1answer
15 views

Convert $\ddot{y}(t) = \dot{y}(t)-y(t)^2$ with $y(0)=y_0, \dot{y}(0) = y_1$ into a first order IVP

We're given the following 2. order IVP $$\ddot{y}(t) = \dot{y}(t)-y(t)^2$$ with initial values $y(0)=y_0, \dot{y}(0) = y_1$. We're asked to convert it into a first Order IVP.
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1answer
16 views

Find out whether the following IVP satisfy the Picard-Lindelöf theorem

We're asked to find out whether the following IVP satisfy the Picard-Lindelöf theorem $\dot{y}(t)= (y(t))^{1/3}, y(0)=1$ $\dot{y}(t)= \sqrt[3]{(y(t)-1)^2}, y(0)=-1$ $\dot{y}(t)= 3 \sqrt[3]{(y(t)-1)^...
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0answers
12 views

Issues for solving a linear partial differential equation of second order

I'm currently solving this partial differential equation: $$u_{xx}+3u_{xy}-4u_{yy}=xy,$$ with $u(x,x)=\sin(x)$ and $\dfrac{\partial }{\partial x} u(x,y) \mid_{y=x} = 0. $ I am just learning how to ...
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0answers
13 views

Help for finding a reference IVP

I have found the following theorem online: If the functions p and g are continuous on the interval $I: \alpha < t < \beta$ containing the point $t = t_0$, then there exists a unique function $y ...
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3answers
52 views

How should I approach this ivp problem with two differentiation about x?

The question given is as followed: $$x dy - y dx - (1-x²)dx = 0, \\ y(1)=1$$ How should I approach this question? I tried to start this problem by finding an integrating factor but what should I ...

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