Solve the separable differential equation ⁠\(4x-6y\sqrt{x^{2}+1}\frac{dy}{dx}=0\) subject to the initial condition ⁠\(y(0)=10\).

We are given the separable differential equation:

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

with the initial condition \( y(0) = 10 \).

### Step 1: Rearrange the equation

First, let's isolate \(\frac{dy}{dx}\) by moving the terms to one side:

\[
6y\sqrt{x^2 + 1} \frac{dy}{dx} = 4x
\]

Now divide both sides by \( 6y\sqrt{x^2 + 1} \) to separate the variables:

\[
\frac{dy}{dx} = \frac{4x}{6y\sqrt{x^2 + 1}} = \frac{2x}{3y\sqrt{x^2 + 1}}
\]

### Step 2: Separate variables

Rearranging the equation to separate \( y \) and \( x \):

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

### Step 3: Integrate both sides

Now integrate both sides.

For the left-hand side:

\[
\int y \, dy = \frac{y^2}{2}
\]

For the right-hand side:

\[
\int \frac{2x}{3\sqrt{x^2 + 1}} \, dx
\]

Let’s perform a substitution for the right-hand side:
- Set \( u = x^2 + 1 \), so \( du = 2x \, dx \).

Thus, the integral becomes:

\[
\int \frac{2x}{3\sqrt{x^2 + 1}} \, dx = \frac{1}{3} \int \frac{du}{\sqrt{u}} = \frac{1}{3} \cdot 2\sqrt{u} = \frac{2}{3}\sqrt{x^2 + 1}
\]

### Step 4: Combine the results

Now we combine the two integrals:

\[
\frac{y^2}{2} = \frac{2}{3}\sqrt{x^2 + 1} + C
\]

Multiply both sides by 2:

\[
y^2 = \frac{4}{3}\sqrt{x^2 + 1} + 2C
\]

Let \( C_1 = 2C \), so:

\[
y^2 = \frac{4}{3}\sqrt{x^2 + 1} + C_1
\]

### Step 5: Apply the initial condition

We know \( y(0) = 10 \), so substitute \( x = 0 \) and \( y = 10 \) into the equation:

\[
10^2 = \frac{4}{3}\sqrt{0^2 + 1} + C_1
\]

This simplifies to:

\[
100 = \frac{4}{3} \times 1 + C_1
\]

\[
100 = \frac{4}{3} + C_1
\]

\[
C_1 = 100 - \frac{4}{3} = \frac{300}{3} - \frac{4}{3} = \frac{296}{3}
\]

### Step 6: Final solution

Substitute \( C_1 \) back into the equation:

\[
y^2 = \frac{4}{3}\sqrt{x^2 + 1} + \frac{296}{3}
\]

Thus, the solution is:

\[
y(x) = \sqrt{\frac{4}{3}\sqrt{x^2 + 1} + \frac{296}{3}}
\]

### Do you have any questions or want more details?

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Here are 5 related questions you might explore:

1. How do you perform separation of variables in differential equations?
2. What is the method of substitution in integrals?
3. How does the initial condition affect the solution of a differential equation?
4. What are separable and non-separable differential equations?
5. Can all first-order differential equations be solved by separation of variables?

**Tip:** Always check the initial conditions after solving a differential equation to ensure the solution is consistent.

Solve the separable differential equation 4x - 6y√(x² + 1) dy/dx = 0 subject to the initial condition y(0) = 10.

Explore how to solve the separable differential equation 4x - 6y√(x² + 1) dy/dx = 0 with a step-by-step solution. Learn how to apply separation of variables, perform substitution in integrals, and use the initial condition y(0) = 10 to find the solution. Suitable for undergraduate calculus students.

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