AP Calculus AB Exam Question:

Consider the differential equation:

(b) Given the initial condition $y(0) = \frac{\pi}{4}$, find the particular solution to the differential equation.

Answer:

(a) To solve the differential equation $\frac{dy}{dx} = x \sin(y)$, we can separate the variables and integrate both sides of the equation.

Starting with the original equation:

To separate the variables, we can multiply both sides of the equation by $\sin(y)$:

We can then divide both sides by $\sin(y)$:

Now, we integrate both sides of the equation:

To integrate $\frac{dy}{\sin(y)}$, we can use the substitution $u = \cos(y)$, which gives us $du = -\sin(y) \, dy$. Therefore, the equation becomes:

Simplifying the integrals leads to:

Substituting back $u = \cos(y)$:

Now, we can solve for $y$:

Exponentiating both sides:

Since $e^{-\frac{1}{2}x^2 - C_1}$ is always positive, we can write it as:

Finally, we can take the inverse cosine of both sides to solve for $y$:

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

where $C_2$ and $n$ are constants.

(b) Given the initial condition $y(0) = \frac{\pi}{4}$, we can substitute $x = 0$ and $y = \frac{\pi}{4}$ into the general solution:

Simplifying the equation:

Since $\frac{\pi}{4}$ is in the range of the inverse cosine function, we can rewrite the equation as:

Simplifying:

Substituting this back into the general solution:

Therefore, the particular solution to the differential equation with the initial condition $y(0) = \frac{\pi}{4}$ is:

where $n$ is an integer.