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The technique of separation of variables is a powerful method used to solve certain types of differential equations. Specifically, it helps when you can express the equation in the form where all instances of one variable are on one side and the other variable is isolated on the opposite side.
This method hinges on the ability to "separate" the equation. For the given differential equation \( \frac{d y}{d x} = 3 \sqrt{x y} \), we can reorganize it so that each side of the equation has only one variable:

• Divide both sides of the equation by \( \sqrt{y} \) to handle the dependency on \( y \).
• Multiply through by \( dx \) to ensure all \( x \)-related terms are together. This gives us \( \frac{d y}{\sqrt{y}} = 3\sqrt{x} \, dx \).

The goal here is straightforward. By doing so, we can easily integrate each side with respect to its respective variable. This isolation of variables means we can process the integration separately, simplifying the problem significantly.

The concept of general solutions in differential equations refers to the expression that encompasses all possible solutions of the equation. In finding a general solution, we usually introduce an arbitrary constant, denoted as \( C \), to account for all potential initial conditions.
For the equation given, after separating variables, we arrive at the stage of integration. Integrating both sides:

• For the \( y \)-side: \( \int \frac{1}{\sqrt{y}} \, dy = 2\sqrt{y} + C_1 \).
• For the \( x \)-side: \( 3 \int \sqrt{x} \, dx = 2 x^{3/2} + C_2 \).

We set the two sides equal, which results as: \( 2\sqrt{y} = 2 x^{3/2} + C \).
This expression forms the implicit general solution of the differential equation. Here, \( C = C_2 - C_1 \) is the constant of integration that provides flexibility for any initial condition the equation might need to satisfy. This constant is essential in defining the specific curve of solutions that meet any required conditions.

Integration techniques are a collection of methods used to evaluate integrals, which are crucial in solving the separated differential equation. In our case, the integration process involves two integrals:

• The left integral: \( \int \frac{1}{\sqrt{y}} \, dy \) is simplified by recognizing it as a standard integral form, resulting in \( 2\sqrt{y} + C_1 \).
• The right integral: \( 3 \int \sqrt{x} \, dx \) also relies on standard integral formulas. Transforming \( \sqrt{x} \) into \( x^{1/2} \) allows us to use the power rule for integrals, yielding \( 2 x^{3/2} + C_2 \).

To solve such problems, recognizing patterns in integrals and applying them efficiently can significantly simplify your work. The constants \( C_1 \) and \( C_2 \) are crucial as they encompass the range of all possible solutions to this differential equation. Overall, integration techniques are indispensable tools in crafting precise and meaningful solutions to differential equations.