91Ó°ÊÓ

Differential equations are equations involving derivatives of a function. In simple terms, they relate a function to its rate of change. Understanding how to solve differential equations is essential for modeling real-world systems, such as motion, growth, decay, and much more. In our example, we start with the differential equation:

• Given: \( y' = \frac{1}{ty + y} \)
• Recognize: It can be rewritten and simplified as \( y' = \frac{1}{y(t+1)} \)

This kind of equation is known as a separable differential equation, which means we can separate the variables on different sides of the equation.

Integration is a fundamental concept in calculus. It is the process of finding the integral, or antiderivative, of a function. When solving differential equations, integration helps us reverse the differentiation process to find the original function. In our example, once we separate the variables, we need to integrate both sides:

• Original simplified form: \( y \, dy = \frac{1}{t+1} \, dt \)
• After separating variables, integrate both: \( \int y \, dy \) and \( \int \frac{1}{t+1} \, dt \)

The integrals we solve here are straightforward:

• \( \int y \, dy = \frac{y^2}{2} \)
• \( \int \frac{1}{t+1} \, dt = \ln|t+1| \)

Remember to add a constant of integration, typically denoted by C.

The variable separation technique is a method used in solving differential equations where we rearrange the given equation in such a way that each variable and its differential are on opposite sides of the equation. This makes it possible to integrate each side independently. For our exercise, this process involves:

• Starting equation: \( y' = \frac{1}{y(t+1)} \)
• Rearrange to separate variables: \( y \, dy = \frac{1}{t+1} \, dt \)

By successfully separating and then integrating both sides, we can solve the differential equation. This brings us to the next step, where we find the integrals and combine them to find the solution in terms of y.

The concepts of differential equations, integration, and variable separation are deeply rooted in calculus. Calculus provides the tools necessary to understand change and motion, making it invaluable in numerous fields such as physics, engineering, economics, and beyond. In our solved problem, we applied calculus methods to progress through the steps:

• First, recognizing the type of differential equation (here, a separable differential equation).
• Next, utilizing the variable separation technique to make the equation integrable.
• Finally, integrating both sides and solving for the variable of interest, y.

Thus, calculus not only helps us in solving mathematical problems but also in interpreting results in a meaningful context. Wrapping up, the solution of our differential equation is expressed as: \( y = \pm \sqrt{2(\ln|t+1| + C)} \). This showcases the entire application of calculus from separation, integration to obtaining the final solution.