Find general solutions (implicit if necessary, explicit if conve-nient) of the differential equations in Problem.
Primes denote derivatives with respect to x. dy/dx = 3(sqrt(xy))

We are given the differential equation:

\[
\frac{dy}{dx} = 3\sqrt{xy}
\]

This is a separable differential equation, which means we can rearrange the terms to separate the variables \(x\) and \(y\). Let's proceed step by step to solve it.

### Step 1: Separate variables
We begin by isolating \(dy\) and \(dx\) terms.

\[
\frac{dy}{dx} = 3\sqrt{xy}
\]

To separate variables, rewrite the square root term:

\[
\frac{dy}{dx} = 3\sqrt{x}\sqrt{y}
\]

Now divide both sides by \(\sqrt{y}\) and multiply by \(dx\):

\[
\frac{dy}{\sqrt{y}} = 3\sqrt{x}dx
\]

### Step 2: Integrate both sides
Now, integrate both sides with respect to their respective variables.

For the left-hand side:

\[
\int \frac{dy}{\sqrt{y}} = 2\sqrt{y}
\]

For the right-hand side:

\[
\int 3\sqrt{x}dx = 3 \cdot \frac{2}{3}x^{3/2} = 2x^{3/2}
\]

### Step 3: Combine and solve for \(y\)
Now that we've integrated, we get:

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

where \(C\) is the constant of integration. Divide by 2:

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

Now square both sides to solve for \(y\):

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

### General Solution:
The general solution to the differential equation is:

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

where \(C\) is an arbitrary constant.

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Do you want any further details on the steps or explanations?

Here are 5 related questions for further exploration:
1. What is the role of the constant \(C\) in the general solution of a differential equation?
2. How do you verify a solution to a differential equation by differentiation?
3. Can you explore the behavior of the solution as \(x \to \infty\)?
4. How would the solution change if the initial condition \(y(x_0) = y_0\) was given?
5. What is the significance of separable differential equations in modeling physical systems?

**Tip:** Always check the domain of your solution, especially when dealing with square roots and other non-linear terms.

Find general solutions (implicit if necessary, explicit if convenient) of the differential equations in Problem. Primes denote derivatives with respect to x. dy/dx = 3(sqrt(xy))

Learn how to find the general solution for the differential equation dy/dx = 3√(xy) using the method of separation of variables. This detailed step-by-step guide includes key formulas and integration techniques. Suitable for undergraduate math students.

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