# 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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### Solution Review: Evaluating Monotonicity of Solutions to IVP

I'm studying a problem from an old ODE exam and I have some general ideas on how to solve it but I feel as though these arguments are kind of heuristically clear but wanting in formality. We're given ...
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### Differential Equation Involving Minimum of Two Functions

This one has completely stumped me. It's from an ODE general exam, I'm given the IVP $$y' = \text{min}(y^2, M),$$ $$y(0) = 1.$$ With $M>1$ and I'm asked to give an explicit solution and discuss ...
• 327
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### IVP with the Banach fixed point theorem: $y' = \sqrt{x} + \sqrt{|y|}$ and $y(0)=0$

I need to use the Banach fixed point theorem to prove that $y' = \sqrt{x} + \sqrt{|y|}$ (for $x \geq 0$) with the initial condition $y(0)=0$ has a unique solution. First of all: y' \...
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### Existence of an unique solution of an ODE without boundary conditions

I would like to ask how to determine if there is a unique solution to an ODE that does not have any boundary conditions, nor initial conditions. It sounds weird but I wanted to know if there is a case ...
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### Second order IVP with a parameter

I am working on this problem. I am studying for an exam. Let $u_{\alpha}$ be the solution of the equation $$u''(t) + F(t)u'(t) + (u(t))^5 - u(t) = 0,$$ with initial ...
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1 vote
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### A second order ordinary differential equation problem

I am trying to solve this problem. I am studying for an exam. Let $F \colon \mathbb{R} \rightarrow \mathbb{R}$ be a $C^1$ function such that $F(1) = 0$. Let $y \colon \mathbb{R} \rightarrow \mathbb{R}$...
• 145
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### Neural network that learns ODE's 'refuses to learn' initial conditions [closed]

I have implemented a simple network that for now i'm just trying to teach the ODE: $$\frac{\text{d}x}{\text{d}t} = x$$ Using the simple code below: ...
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### Prove existence of solution for IVP

I am trying to solve this ODE problem. I am studying for an exam. Let $f \colon \mathbb{R}^n \rightarrow \mathbb{R}^n$ be $C^1$ such that $|f(x)| \leq 1 + |x|^{\alpha}$. Consider the IVP \begin{...
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### Equivalence between IVP and integral equation

I was trying to prove the following theorem: If $f(x, y)$ is continuous on some region $R \subseteq \mathbb R^2$ then any solution of IVP $$y'(x)=f(x,y(x)), y(x_0)=y_0 \tag{1}$$ is also a solution of ...
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### Solution of an IVP through Laplace transform

Let $𝑦(𝑡)$ be the solution of the initial value problem $$y''+4y=\begin{cases} t, & 0\leq t\leq 2\\ 2, & 2<t<\infty \end{cases}.$$ Also, it is given that $$y(0)=0, y'(0)=0.$$ Given ...
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### Existence of complex solutions to real IVP

Consider the IVP: $$y'(x)=f(x,y(x))\\y(x_0)=y_0$$ It is known that: If $f$ is continuous and Lipschitz in 2nd variable then: Existence and Uniqueness If $f$ is continuous then: just Existence Now ...
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### Solve the initial value problem $y'=e^{-y^2}+1, \, y(0)=0$

Consider the ODE(ordinary differential equation) $$y'=e^{-y^2}+1, y(0)=0.$$ Which of the following statements are true for given ODE? $P:$ The ODE has unique solution on $\mathbb R.$ $Q:$ The ...
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### Show that there's an interval $I \subseteq \mathbb{R}$ with $3 \in I$, so that the initial value problem has a unique solution.

Show that there's an interval $I \subseteq \mathbb{R}$ with $3 \in I$, so that the initial value problem $$x'(t) = (x(t) - t)e^{t^2+x(t)} \text{ with } x(3) = 7$$ has a unique solution. So far I have ...
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### Initial value problem involving hyperbolic function.

Let $$y(t):[0, \infty)\to \mathbb{R}$$ be the solution of the problem $$y'(t)=(1-y^2(t))\cosh y(t)\ \text{for}\ t>0, \ y(0)=y_0.$$ Find the set of real values of $y_0$ such that $y(t)$ mentioned ...
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### Solutions to $y'=y^\alpha$ vary continuously in neighbourhood

Consider the differential equation $y' = y^\alpha$ with initial condition $y(0) = c_0 > 0$. Let $\alpha$ vary in a neighborhood of $1 \in \mathbb{R}$. Of course, for $\alpha = 1$ we get the ...
1 vote
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### Flow is defined using a $C^1$ function is $C^1$

I have a flow defined by the initial value problem: $$\frac{d}{dt}y_t(x)=f_t(y_t(x)), \quad y_0(x)=x$$ where $f_t:\mathbb{R}^k\rightarrow\mathbb{R}^k$. I know the above problem has a unique solution ...
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### Find a power series solution of the Cauchy-Euler Equation around $x=0$

I have the following problem: Find a power series solution of the Cauchy-Euler Equation around $x=0$: $$(2x+5)y'' + y'= 0, \quad y(0)=2, \quad y'(0)=3.$$ When I tried solving it, the recurrence I got ...
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### proving an ODE with a specific solution can't exist using the existence and uniqueness theorem

I'm having trouble completing the following exercise from my homework: Let $y'' = f(x, y, y')$ be an ODE that fulfills the criteria for the existence and uniqueness theorem (Picard–Lindelöf theorem) ...
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