91Ó°ÊÓ

A separable differential equation is a type of differential equation where the variables can be separated on different sides of the equation. This means you can organize all terms involving one variable (like \(y\)) on one side, and all terms involving another variable (like \(x\)) on the other. The goal is to make the equation easier to solve by focusing on one variable at a time.

In the exercise, the original equation \(x y y' = x^2 + 1\) was rewritten to separate the variables: \(y' = \frac{x^2 + 1}{xy}\). This allows us to rearrange terms to eventually arrive at \(y \, dy = \left( x + \frac{1}{x} \right) dx\). Each side contains only one variable, making it ready for integration, a key step in solving separable differential equations.

Once you have a separable differential equation, the next step typically involves integrating both sides of the equation. Integration is a process that helps find a function when its derivative is known.

In our case, after separating the variables, we obtained \(\int y \, dy = \int \left( x + \frac{1}{x} \right) dx\). Here, you need to integrate the \(y\) terms with respect to \(y\) and the \(x\) terms with respect to \(x\).

• The integration of \(y\) gives \(\frac{y^2}{2}\).
• The integration for \(x\) includes two parts: \(\int x \, dx = \frac{x^2}{2}\) and \(\int \frac{1}{x} \, dx = \ln|x|\).

The result is a general form involving a constant \(C\), which we denote as the integration constant. These integration techniques are crucial tools when working with differential equations.

The general solution of a differential equation represents all possible solutions, given by the integration process, typically involving an arbitrary constant known as the integration constant.

After solving the differential equation, we arrived at \(\frac{y^2}{2} = \frac{x^2}{2} + \ln|x| + C\). To simplify, multiply by 2, resulting in \(y^2 = x^2 + 2 \ln|x| + 2C\). Solving for \(y\) gives us the general solution: \(y = \pm \sqrt{x^2 + 2 \ln|x| + C}\).

This expression includes all possible solutions for the differential equation, although only specific solutions are valid based on initial conditions supplied or additional constraints, such as domain restrictions.

The integration constant is crucial when finding the general solution to a differential equation. It accounts for the family of solutions due to the indefinite integration process.

During integration, each side was assigned a constant since the original boundaries or initial conditions are unknown. Here, it appeared as \(C\) in \(\int y \, dy = \int \left( x + \frac{1}{x} \right) dx + C\). After simplification of the general solution, the integration constant was adjusted through multiplication by 2, but it's important to note that the constant \(C\) remains arbitrary until further specified by additional conditions.

• This constant allows for flexibility in solutions and adapts when initial values are introduced.
• This concept ensures that the solution is adaptable to changes, representing a broad range of potential solutions.

Understanding the integration constant is vital in differential equations for handling various scenarios that satisfy the initial differential equation.