91Ó°ÊÓ

Differential equations come in many forms, each with its own methods for finding solutions. The equation \( \frac{d y}{d x} = \frac{1}{x^2} \) is known to be a first-order separable equation. Essentially, these equations allow us to separate variables in order to solve them. This means we can manipulate the equation so that every instance of one variable, such as \( y \), is on one side, while every instance of the other variable, like \( x \), is on the other.

This approach simplifies the process of finding a solution by turning the differential equation into two simpler one-dimensional integrals. Here's a simple way to identify if a differential equation is first-order and separable:
- Look for the derivative of a function \( \frac{d y}{d x} \) expressed as a multiplication or division of separate functions of \( x \) and \( y \).
- Try to rearrange the terms so that it looks like \( g(y) dy = h(x) dx \).

Once a differential equation is in this form, you are set to move on to the next step: integration.

The solution of a differential equation isn’t complete without applying initial conditions. An initial condition is an additional piece of information that allows us to find a unique solution for our equation among an infinite number of possible solutions. In the given exercise, the initial condition is \( y(1) = 5 \).

By applying initial conditions, you determine specific values for any constants integrated into the solution. Here’s a step-by-step thought process:
- Calculate the general solution of the differential equation first.
- Substitute the initial values provided into this general solution.
- Solve for the constant(s) using these substitute values.

This process turns the general solution into a particular solution, which is crucial for truly understanding the behavior of the differential equation over time.

Integration is a key tool in solving differential equations, particularly first-order separable ones. The process involves finding antiderivatives of separated functions. Once the variables are successfully separated, as in \( dy = \frac{1}{x^2} dx \), integration is required to find the function solution.

The steps for integration here are:

• Integrate both sides separately.
• The left side \( \int dy \) integrates to \( y \).
• The right side \( \int \frac{1}{x^2} dx \) evaluates to \( -\frac{1}{x} \) plus a constant \( C \).

This step highlights the importance not only of integration itself but of constants of integration. The constant \( C \) is critically important because it allows adjustments based on initial conditions to achieve one specific solution from the general form.

Once you have separated variables and integrated both sides of the equation, you achieve what is known as the general solution. In our example, it is \( y = -\frac{1}{x} + C \). To find a specific answer, integrate again with the initial condition, which in this exercise is \( y(1) = 5 \).

The found solution captures the specific behavior of the described system at the given starting point. Here is how you complete the function solution:

• Plug the initial conditions into the general solution.
• Substitute \( y = 5 \) and \( x = 1 \) into the equation: \( 5 = -1 + C \).
• Solve for \( C \), giving us a value of \( C = 6 \).

With the calculated \( C \), the function solution becomes finalized: \( y = -\frac{1}{x} + 6 \). This solution is now finely-tuned to the original problem, accurately reflecting the unique path of the function based on the initial condition provided.