First Order ODE

First order ordinary differential equations have the general form of

y' = f(x, y),

or

M(x, y)dx + N(x, y)dy = 0.

Although the above general forms look simple, there is no single rule to solve them.Before trying to find a solution of such an equation, it is useful to know whether a solution exists.

If F is a function of two variables that is continuous at (x₀, y₀) then there exists a number a > 0 and a continuously differentiable function y defined on the interval (x₀ - a, x₀ + a) such that y(x₀) = y₀ and y'(x) = F(x, y(x)) for all x in the interval. If the partial derivative of F with respect to y is continuous on an open rectangle containing (x₀, y₀) then there exists a > 0 such that the initial value problem has a unique solution on the interval (x₀ - a, x₀ + a).

The condition guaranteeing a unique solution (that the partial derivative of F with respect to y be continuous) is relatively mild, and is satisfied in almost all the examples we study. After looking at some direction fields, you might wonder how even an initial value problem that does not satisfy the condition could have more than one solution. Here is an example.

Example 1

Consider the initial value problem
y'(x) = (y(x))^1/2
y(0) = 0.

This problem does not satisfy the condition in the proposition for a unique solution, because the square root function is not differentiable at 0.

The problem has two solutions: y(x) = 0 for all x, and y(x) = (2x/3)^3/2 for all x.