Let's solve the given separable differential equation:

The equation:

$4x - 6y\sqrt{x^2 + 1} \frac{dy}{dx} = 0$

Step 1: Rearrange the equation

Reorganize terms to separate $x$ and $y$: $\frac{dy}{dx} = \frac{4x}{6y\sqrt{x^2 + 1}}$

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

Step 2: Rewrite in separable form

Rearrange to separate $y$ and $x$: $y \, dy = \frac{2x}{3\sqrt{x^2 + 1}} \, dx$

Step 3: Integrate both sides

• Left-hand side: $\int y \, dy = \frac{y^2}{2}$

• Right-hand side: Substitute $u = x^2 + 1$, so $du = 2x \, dx$. The integral becomes: $\int \frac{2x}{3\sqrt{x^2 + 1}} \, dx = \frac{1}{3} \int u^{-1/2} \, du = \frac{1}{3} \cdot 2u^{1/2} = \frac{2}{3}\sqrt{x^2 + 1}$

Step 4: Combine results

Equating the results of the integrals: $\frac{y^2}{2} = \frac{2}{3}\sqrt{x^2 + 1} + C$

Multiply through by 2: $y^2 = \frac{4}{3}\sqrt{x^2 + 1} + C_1$

Step 5: Solve for $y$

$y = \pm\sqrt{\frac{4}{3}\sqrt{x^2 + 1} + C_1}$

Step 6: Use the initial condition $y(0) = 1$

Substitute $x = 0$ and $y = 1$: $1 = \sqrt{\frac{4}{3}\sqrt{0^2 + 1} + C_1}$

$1 = \sqrt{\frac{4}{3} \cdot 1 + C_1}$

$1 = \sqrt{\frac{4}{3} + C_1}$

Square both sides: $1 = \frac{4}{3} + C_1$

Solve for $C_1$: $C_1 = 1 - \frac{4}{3} = -\frac{1}{3}$

Step 7: Final solution

Substitute $C_1 = -\frac{1}{3}$ into the equation: $y = \pm\sqrt{\frac{4}{3}\sqrt{x^2 + 1} - \frac{1}{3}}$

Since $y(0) = 1 > 0$, take the positive root: $y = \sqrt{\frac{4}{3}\sqrt{x^2 + 1} - \frac{1}{3}}$

Would you like more details on any step? Here are some related questions to deepen your understanding:

1. How do you determine when a differential equation is separable?
2. What are some common techniques for solving non-separable differential equations?
3. Why is substitution helpful in simplifying integrals?
4. How does the initial condition affect the constant of integration?
5. How can you verify the solution to a differential equation?

Tip: Always verify that your final solution satisfies both the differential equation and the initial condition to ensure accuracy.