# 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.

688 questions
Filter by
Sorted by
Tagged with
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 ...
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 ...
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 $x = 0$ ...
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 ...
14 views

11 views

27 views

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, ...
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. ...
63 views

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 ...
38 views

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, ...
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 ...
25 views

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 ...
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 ...
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.
We're asked to find out whether the following IVP satisfy the Picard-Lindelöf theorem $\dot{y}(t)= (y(t))^{1/3}, y(0)=1$ $\dot{y}(t)= \sqrt{(y(t)-1)^2}, y(0)=-1$ $\dot{y}(t)= 3 \sqrt{(y(t)-1)^... 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. $I am just learning how to ... 0answers 13 views ### Help for finding a reference IVP I have found the following theorem online: If the functions p and g are continuous on the interval$I: \alpha < t < \beta$containing the point$t = t_0$, then there exists a unique function$y ...
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 ...