91Ó°ÊÓ

To solve a differential equation using the method of separation of variables, we aim to rearrange the equation so that all terms involving the dependent variable (e.g., \( y \)) are on one side, and all terms involving the independent variable (e.g., \( x \)) are on the other side. This forms an equation that can be solved by integrating both sides independently. In the given exercise, this involves rewriting the differential equation from \( \sqrt{x^{2}-9} y^{\prime}=5 x \) to \( y^{\prime} = \frac{5x}{\sqrt{x^{2}-9}} \).

• Move terms involving \( y \) and its derivative \( y' \) to one side.
• Move terms involving \( x \) to the opposite side.

Once arranged correctly, the equation becomes separable. This step is crucial because it simplifies the process by allowing us to apply integration techniques separately to both sides.

Integration is the next critical step after separating the variables. The aim is to find an antiderivative or integral of both sides of the equation. Here, the integral we need to solve is \( y = \int \frac{5x}{\sqrt{x^{2}-9}} \, dx \). This requires identifying a suitable technique to evaluate the integral, such as:

• Substitution: Frequently used when an expression's derivative is present in the integrand.
• Integration by parts: Useful for products of functions.
• Partial fractions: Effective for decomposing rational expressions.

For this specific problem, substitution is often the best approach. You might substitute \( u = x^{2} - 9 \), which simplifies the integration. Performing such techniques can help in finding the antiderivative needed for the solution.

In differential equations, the general solution encompasses all possible solutions to the equation. It typically includes arbitrary constants, as these reflect the family's range of functions that solve the equation. In our exercise, the general solution is derived as \( y = 5(\sqrt{x^{2}-9} + C) \), where \( C \) is the constant of integration. This constant emerges from the integration process and symbolizes any particular solution fitting a specific initial condition.

• The constant \( C \) allows the solution to adapt to various initial values.
• Finding a particular solution involves specifying \( C \) based on additional information, such as an initial condition.

Understanding the role of the general solution and the constant \( C \) is crucial for recognizing how differential equations can apply to multiple scenarios and conditions.