Solve the initial value problem 

StartFraction dy Over dx EndFractiondydxequals=StartRoot 3 plus sine x EndRoot left parenthesis 1 plus y squared right parenthesis3+sinx1+y2​, 

with 

​y(0)equals=1. 

Use t as the variable of integration in the explicit solution. 


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.

---

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.

Solve the initial value problem dy/dx = √(3 + sin(x)) (1 + y^2), with y(0) = 1. Use t as the variable of integration in the explicit solution.

Learn how to solve the initial value problem dy/dx = √(3 + sin(x)) (1 + y^2), with the initial condition y(0) = 1. This solution covers the method of separation of variables, integration, and determining the constant of integration. Suitable for advanced high school or early undergraduate students.

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