The general solution is
Example 8 __
Solve the d.e. , given the initial condition y(0) = 2.
SOLUTION:
We rewrite the equation as y dy = −x dx. We then integrate, getting
Since y(0) = 2, we get 4 + 0 = C; the particular solution is therefore x^2 + y^2 = 4.
(We need to specify above that y > 0. Why?)
Example 9 __
If and t = 0 when s = 1, find s when t = 9.
SOLUTION:
We separate variables: