Use separation of variables to find the general solution of the differential equation. ex(y + 2) = 1
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.
To solve the differential equation using separation of variables, we need to rearrange it and find a way to separate the variables on either side of the equation.
The given differential equation is ex(y + 2) = 1.
Let's begin by dividing both sides of the equation by ex:
(y + 2) = 1/ex
Now, we can separate the variables by moving the terms involving y to one side and the terms involving ex to the other side:
(y + 2)dy = 1/ex dx
Next, we can integrate both sides of the equation with respect to their corresponding variables:
∫(y + 2)dy = ∫1/ex dx
Integrating the left side gives:
(y^2/2 + 2y) = ∫1/ex dx
To integrate the right side, we can use the integral of e^x, which is e^x:
(y^2/2 + 2y) = e^x + C
where C is the constant of integration.
Note that we usually write the constant of integration on the right side, but it can be placed on either side.
Finally, we can write the general solution by isolating y:
y^2/2 + 2y = e^x + C
This is the general solution to the given differential equation.