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.

E 1.2 Exercises