AP Calculus AB Exam Question:

Consider the following differential equation:

a) Apply the separation of variables technique to solve the differential equation.

b) Determine the particular solution that satisfies the initial condition $y(\frac{\pi}{2}) = 3$.

Solution:

a) To solve the given differential equation using separation of variables, we first write it in the form:

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

On the left side, we obtain:

To integrate the right side, we use integration by parts with $u = \sin(2x)$ and $dv = \frac{1}{x} \cdot dx$. This gives us:

Simplifying further:

The integral on the right side can be evaluated using integration by parts as well. Setting $u = \cos(2x)$ and $dv = \frac{1}{x} \cdot dx$, we have:

We substitute this result back into our previous equation:

We now have an equation involving the integral of $\frac{1}{x} \cdot \sin(2x)$. Rearranging this equation, we obtain:

Substituting this back into the initial integration equation, we have:

Simplifying further, we obtain:

Finally, we exponentiate both sides to eliminate the natural logarithm:

And since $|y|$ represents the absolute value of $y$, we can remove the absolute value sign:

Therefore, the general solution to the given differential equation is:

where $C$ is an arbitrary constant.

b) To determine the particular solution that satisfies the initial condition $y(\frac{\pi}{2}) = 3$, we substitute $\frac{\pi}{2}$ for $x$ and $3$ for $y$ in the general solution:

Simplifying further:

Since $\cos(\pi) = -1$ and $\sin(\pi) = 0$, we have:

Simplifying further:

To solve for $C$, we divide both sides of the equation by $e^{\frac{1}{2}}$:

Therefore, the particular solution that satisfies the initial condition $y(\frac{\pi}{2}) = 3$ is: