The document outlines the solution to the initial value problem represented by the equation 19. sin(2x)dx + cos(3y)dy = 0 with the initial condition y(pi/2) = pi/3. By separating variables and integrating, the implicit solution is derived as -3cos(2x) + 2sin(3y) = 3. The constant 'c' is determined using the initial condition, ultimately providing the implicit form of the solution.

1. Find the solution of the given initial value problem in explicit form:
19. sin(2x)dx + cos(3y)dy = 0, y(pi/2) = pi/3
Solution
sin(2x)dx + cos(3y)dy = 0
As variables are separate integrate to find x and y
-cos 2x/2 +sin3y/3 =C
Or -3cos2x+2sin3y = C'
x=pi/2, y =pi/3
Substitute to get
-3cospi+2sinpi = C' = 3
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Hence solution is the implicit equation
-3cos2x+2sin3y = 3