AP Calculus AB Exam Question - Separation of Variables:

Find the general solution to the following differential equation using separation of variables:

Solution:

To solve this differential equation, we will use the method of separation of variables. The first step is to write the equation in the form $\frac{dy}{dx} = g(x) \cdot f(y)$, where $g(x)$ represents the function of $x$ and $f(y)$ represents the function of $y$.

For our given equation, we can rewrite it as:

Let's multiply both sides by $dx$ to separate the variables:

Now, we separate the variables by moving $dx$ to one side and $dy$ to the other side:

Next, we integrate both sides of the equation with respect to their respective variables:

The integral of $dy$ is simply $y$, and we can evaluate the integral of $\frac{1}{x-1}$ using the natural logarithm function:

To evaluate this integral, we use the substitution method. Let $u = x-1$, then $du = dx$. Substituting $u$ and $du$ into the equation, we have:

Integrating $\frac{1}{u}$ with respect to $u$ gives us $\ln|u| + C$, where $C$ is the constant of integration:

Finally, we have the general solution to the differential equation:

where $C$ is the constant of integration.