Section 2.4
Linear vs. Non-Linear Equations

Existence and Uniqueness Theorem for general first-order equations

Theorem: Let the functions f and $\frac{\partial f}{\partial y}$ be continuous in some rectangle $\alpha < t < \beta, \gamma < y < \delta$ containing the point (t₀,y₀). Then, in some interval t₀ - h < t < t₀ + h contained in $\alpha < t < \beta$, there is a unique solution $y = \phi(t)$ of the initial value problem

$$y' = f(t,y),\hspace{1in} y(t_0)=y_0$$

Other differences found when dealing with non-linear equations:

$\clubsuit$ Even if a solution is found, it may be difficult to determine the interval where the solution is valid

$\diamondsuit$ A general solution may not provide every solution

$\heartsuit$ Solutions found are often implicit rather than explicit

$\spadesuit$ Numerical approximations are of great importance