Find a function y = f(x) satisfying the given differential equation and the prescribed initial condition Dy/dx = 6/(x^2 + 4), y(0) = 0 Give me the general solution Y(x) = _____? and the solution to the initial value problem Y(x)= _____?

To solve the given differential equation with the initial condition, we can follow these steps:

### Given Differential Equation:
\[
\frac{dy}{dx} = \frac{6}{x^2 + 4}
\]

### Step 1: Find the General Solution
To find the general solution \( y = f(x) \), we integrate both sides of the equation with respect to \( x \):

\[
y = \int \frac{6}{x^2 + 4} \, dx
\]

We can simplify the integral by recognizing that \( x^2 + 4 \) can be rewritten as:

\[
x^2 + 4 = x^2 + 2^2
\]

This suggests that we can use the standard formula for the integral of \( \frac{1}{x^2 + a^2} \):

\[
\int \frac{1}{x^2 + a^2} \, dx = \frac{1}{a} \arctan\left(\frac{x}{a}\right)
\]

In our case, \( a = 2 \), so the integral becomes:

\[
y = 6 \times \frac{1}{2} \arctan\left(\frac{x}{2}\right) + C
\]

Simplifying, we get:

\[
y = 3 \arctan\left(\frac{x}{2}\right) + C
\]

So, the **general solution** is:

\[
y(x) = 3 \arctan\left(\frac{x}{2}\right) + C
\]

### Step 2: Apply the Initial Condition
We are given the initial condition \( y(0) = 0 \). Substituting \( x = 0 \) and \( y = 0 \) into the general solution, we get:

\[
0 = 3 \arctan\left(\frac{0}{2}\right) + C
\]

Since \( \arctan(0) = 0 \), this simplifies to:

\[
0 = 0 + C \implies C = 0
\]

Thus, the **solution to the initial value problem** is:

\[
y(x) = 3 \arctan\left(\frac{x}{2}\right)
\]

### Final Answers:

1. **General Solution:**
   \[
   y(x) = 3 \arctan\left(\frac{x}{2}\right) + C
   \]

2. **Solution to the Initial Value Problem:**
   \[
   y(x) = 3 \arctan\left(\frac{x}{2}\right)
   \]

Would you like details on any step, or do you have any questions?

### Related Questions:

1. What is the formula for the integral of \( \frac{1}{x^2 + a^2} \)?
2. How do you find the particular solution from a general solution using initial conditions?
3. What is the significance of the constant \( C \) in the general solution?
4. How does the \( \arctan \) function behave as \( x \) approaches infinity?
5. Can you solve a similar differential equation with a different initial condition?
6. What if the differential equation had a different denominator, such as \( x^2 + 9 \)?
7. How does substitution help in integrating functions like \( \frac{1}{x^2 + 4} \)?
8. What other methods can be used to solve differential equations?

### Tip:
When solving differential equations, always check your solution by differentiating it and verifying that it satisfies the original equation.

Explore how to solve the differential equation Dy/dx = 6/(x^2 + 4) with the initial condition y(0) = 0. Learn the step-by-step process, including finding the general solution and applying the initial condition. Suitable for advanced college-level mathematics.

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