Lecture Notes in Differential Equations - Bruce E. Shapiro

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Figure 1.1 illustrates what this means. The initial condition is given by
the point (t 0 , y 0 ) (the horizontal axis is the t-axis; the vertical axis is y). If
there is some number M such that |∂f/∂y| < M everywhere in the box R,
then there is some region N where we can draw the curve through (t 0 , y 0 ).
This curve is the solution of the IVP. 3
Figure 1.1: Illustration of the fundamental existence theorem. If f is
bounded, continuous and differentiable in some neighborhood R of (t 0 , R 0 ),
and the partial derivative ∂f∂y is also bounded, then there is some (possibly
smaller) neighborhood of (t 0 , R 0 ) through which a unique solution to
the initial value problem, with the solution passing through (t 0 , y 0 ), exists.
This does not mean we are able to find a formula for the solution.
A solution may be either implicit or explicit. A solution y = φ(t) is said
to be explicit if the dependent variable (y in this case) can be written
explicitly as a function of the independent variable (t, in this case). A
relationship F (t, y) = 0 is said to represent and implicit solution of the
differential equation on some interval I if there some function φ(t) such
that F (t, φ(t)) = 0 and the relationship F (t, y) = 0 satisfies the differential
equation. For example, equation 1.34 represents the solution of {dy/dt =
t/y, y(1) = 2} implicitly on the interval I = [− √ 3, √ 3] which (1.35) is an
explicit solution of the same initial value problem problem.
3 We also require that |f| < M everywhere on R and that f be continuous and
differentiable.