We are given the following initial value problem (IVP):

$\frac{dy}{dx} = \sqrt{3 + \sin x}(1 + y^2), \quad y(0) = 1$

To solve this IVP, we will use the method of separation of variables. Let's break it down step by step:

Step 1: Separate the variables

We can rewrite the given differential equation as:

$\frac{dy}{1 + y^2} = \sqrt{3 + \sin x} \, dx$

Step 2: Integrate both sides

Now, we will integrate both sides.

1. The left side involves the integral of $\frac{1}{1 + y^2}$, which is a standard integral:

$\int \frac{dy}{1 + y^2} = \tan^{-1}(y)$

2. The right side involves the integral of $\sqrt{3 + \sin x}$, and we are asked to use $t$ as the variable of integration. So, we will rewrite $x$ as $t$ and integrate:

$\int \sqrt{3 + \sin t} \, dt$

At this point, we cannot find a simple closed-form expression for the integral of $\sqrt{3 + \sin t}$, so we will leave it in integral form for now.

Thus, after integrating, we have:

$\tan^{-1}(y) = \int \sqrt{3 + \sin t} \, dt + C$

Step 3: Solve for the constant $C$

To find the constant of integration $C$, we use the initial condition $y(0) = 1$. Plugging this into the equation:

$\tan^{-1}(1) = \int \sqrt{3 + \sin 0} \, dt + C$

Since $\tan^{-1}(1) = \frac{\pi}{4}$ and $\sin(0) = 0$, the equation becomes:

$\frac{\pi}{4} = \int \sqrt{3} \, dt \Big|_{t=0} + C$

The integral of a constant $\sqrt{3}$ is simply $\sqrt{3} t$, so:

$\frac{\pi}{4} = \sqrt{3} \cdot 0 + C$

Thus, $C = \frac{\pi}{4}$.

Step 4: Final solution

Now we substitute $C$ back into the equation:

$\tan^{-1}(y) = \int \sqrt{3 + \sin t} \, dt + \frac{\pi}{4}$

Finally, solving for $y$:

$y = \tan \left( \int \sqrt{3 + \sin t} \, dt + \frac{\pi}{4} \right)$

This is the explicit solution to the initial value problem.

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Would you like more details or have any questions about the steps?

Related Questions:

1. How do we compute the integral of $\sqrt{3 + \sin t}$ in closed form?
2. What is the geometrical interpretation of separating variables in differential equations?
3. Can we use numerical methods to solve the integral of $\sqrt{3 + \sin t}$?
4. What are some examples of initial value problems that cannot be solved analytically?
5. How does the initial condition affect the solution to a differential equation?

Tip:

When separating variables in differential equations, always check if both sides can be integrated directly or if a substitution is needed to simplify the process.