# 1.2 | Solutions and Initial Value Problems

A function $$\phi(x)$$ that when substituted for $$y$$ in
$F\left(x,y,\frac{dy}{dx}, \ldots, \frac{d^ny}{dx^n} \right) = 0,$
satisfies the equation for all $$x$$ in the interval $$I$$ is called an explicit solution to the equation on $$I$$.
A relation $$G(x,y)=0$$ is said to be an implicit solution to
$F\left(x,y,\frac{dy}{dx}, \ldots, \frac{d^ny}{dx^n} \right) = 0,$
on the interval $$I$$ if it defines one or more explicit solutions on $$I$$.
By an initial value problem for an $$n$$th-order differential equation
$F\left(x,y,\frac{dy}{dx}, \ldots, \frac{d^ny}{dx^n} \right) = 0,$
we mean: Find a solution to the differential equation on an interval $$I$$ that satisfies $$x_0$$ the $$n$$ initial conditions
\begin{aligned} y(x_0) & = y_0, \\ \frac{dy}{dx}(x_0) & = y_1, \\ & \vdots \\ \frac{d^{n-1}y}{dx^{n-1}}(x_0) & = y_{n-1}, \end{aligned}
where $$x_0 \in I$$ and $$y_0, y_1, \ldots, y_{n-1}$$ are given constants.
Consider the initial value problem
$\frac{dy}{dx} = f(x,y), \quad y(x_0) = y_0.$
If $$f$$ and $$\partial f/ \partial y$$ are continuous functions in some rectangle
$R = \left \lbrace (x,y) \; | \; a < x< b, \; c < y < d \right \rbrace$ that contains the point $$(x_0, y_0)$$, then the initial value problem has a unique solution $$\phi(x)$$ in some interval $$x_0 - \delta < x < x_0 + \delta,$$ where $$\delta$$ is a positive number.