To use separation of variables, we need to rewrite the equation in the form dy/dx = f(x)g(y), where f(x) and g(y) are functions of x and y, respectively.
Starting with ex(y + 2) = 1, we can first divide both sides by e^(y+2) to obtain:
dy/dx = -1/e^(y+2)
Next, we can separate the variables by multiplying both sides by e^(y+2) and dx, giving:
e^(y+2) dy = -dx
Finally, we can integrate both sides with respect to their respective variables:
∫ e^(y+2) dy = ∫ -dx
Simplifying the integrals:
e^(y+2)/1 = -x + C
where C is the constant of integration.
Solving for y:
y+2 = ln(-x+C)
y = ln(-x+C) - 2
Therefore, the general solution of the differential equation ex(y + 2) = 1 is y = ln(-x+C) - 2, where C is an arbitrary constant.
Use separation of variables to find the general solution of the differential equation. ex(y + 2) = 1
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