AP Calculus AB Exam Question - Separation of Variables

Consider the following differential equation:

(b) Use the solution from part (a) to find the particular solution that satisfies the initial condition $y(0) = 2$.

Solution:

(a) To solve the differential equation using separation of variables, we need to write it in the form $\frac{dy}{dx} = g(x)h(y)$, where $g(x)$ is a function of $x$ only and $h(y)$ is a function of $y$ only.

Given equation: $\frac{dy}{dx} = xy$

So, let's rewrite it as:

Now, we can separate the variables:

Integrating both sides, we get:

Here, $C_1$ represents an arbitrary constant.

To get rid of the absolute value, we can rewrite it as:

Introducing another arbitrary constant, $C_2 = \pm e^{C_1}$, we have:

So, the solution to the given differential equation is:

Therefore, the particular solution is:

And we're done!