then integration yields
Example 10 __
If (ln y) , and y = e when x = 1, find the value of y greater than 1 that
corresponds to x = e^4.
SOLUTION:
Separating, we get . We integrate:
Using y = e when x = 1, we have , or . This is true only if C = 1,
and we choose the positive square root. Hence, . When x = e^4 ,
.
Example 11 __
Find the general solution of the differential equation .
SOLUTION:
We rewrite.
Taking antiderivatives yields eu = ev + C, or u = ln(ev + c).