# 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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### Uniqueness of solution of specific ODE

I'm trying to solve the following problem. It's exercise 1.6 of the book Ordinary Differential Equations by Barreira & Valls. Let $f:(a,b) \rightarrow \mathbb{R} \backslash \{0\}$ be a continuous ...
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### Statement about inequalities of two distinct solutions of an ODE

I have the following initial values problem: \begin{equation} \left\{\begin{array}{@{}l@{}} y'(t)= t\tan(y(t) - \frac{\pi}{4})\\ y(0) = \frac{\pi}{2} \end{array}\right.\,. \end{equation} ...
### Solve initial value problem $\frac{dy}{dx}=x^2(1+y), y(0)=3$
We have $\int\frac{1}{1+y}dy=\int{x^2dx}$, $\ln|1+y|=\frac{x^3}3+C$ $C=\ln4$ The solution from my textbook is $y=4e^{\frac{x^3}3}-1$, which implies that $1+y$ is always positive. Why is it positive?