Question

dy/dx=x^2+2x+a where a is a constant. given that, y=18 at x=3 and y=6 at x=-3, find an expression for y in terms of x.

Answer 1

dy/dx = x² + 2x + a
∫dy =∫(x² + 2x + a)dx
inegrate both sides,
∫dy = ∫x².dx+ 2∫x.dx + a∫dx
y = x³/3 + x² + ax + C

now, x = 3 then y = 18
18 = (3)³/3 + (3)² +3a + C
3a + C = 0 ________________(1)

again,
x = -3 then y = 6
6 = (-3)³/3 + (-3)² -3a + C
6 = -9 + 9 -3a + C
-3a + C = 6__________________(2)
solve eqns (1) and (2)

C = 3 and a = -1

hence,
y = x³/3 + x² -x + 3