2. SEPARABLE EQUATIONS
dy 6x5 2x + 1
(4)
=
= (6x5
y
dx
cos y + e
2x + 1) ·
1
. We have
cos y + ey
(cos y + ey ) dy = (6x5
Then
Z
(cos y + ey ) dy =
Z
(6x5
31
2x + 1) dx.
2x + 1) dx + K =)
y
6
2
sin
| y + e = x{z x + x + K}
implicit solution
We check using implicit di↵erentiation:
(cos y)y 0 + ey y 0 = 6x5
y 0(cos y + ey ) = 6x5
2x + 1 =)
2x + 1 =)
dy
6x5 2x + 1
0
=y =
dx
cos y + ey
Indefinite Integral Convention for Di↵erential Equations
Z
The indefinite integral g(x) dx represernts any (single) antiderivative of g(x).
In other words, it means any function G(x) with the derivative
d
G(x) = g(x).
dx