91Ó°ÊÓ

When solving first-order ordinary differential equations, integration techniques are essential. These techniques help us find the function that satisfies a given differential equation. In this exercise, you have the differential equation: \( y' = \frac{t}{1+t^2} \). To solve this, we integrate the right-hand side with respect to \( t \).

This specific integral, \( \int \frac{t}{1+t^2} \, dt \), involves using a substitution method, making it easier to simplify and solve. By understanding the right integration technique, you can transform complex-looking integrals into more approachable forms. The end goal is always to express the solution in a simpler form, such as \( y(t) = \text{integrated expression} + C \). The constant \( C \) holds the initial conditions if they are provided.

The substitution method is a powerful tool in integration, especially when dealing with complex expressions. Here, substitution allows a more straightforward integration process. For the integral \( \int \frac{t}{1+t^2} \, dt \), we set \( u = 1 + t^2 \).

Upon differentiating, we find \( du = 2t \, dt \). This expression helps us rearrange the integral into \( \int \frac{1}{2u} \, du \). This newly formed integral is simpler, as it involves expressions in \( u \) rather than \( t \).

• The substitution method often involves a few key steps:
• Choose substitution variables that simplify the original problem, like \( u = 1 + t^2 \).
• Express \( dt \) or other parts of the integrand in terms of \( du \).
• Transform the integrand into a simpler integral.

After integration, we substitute back the original variables to find our solution.

The natural logarithm appears often in integration, especially when dealing with integrals of the form \( \int \frac{1}{u} \, du \). In this scenario, using the substitution method simplified our task to \( \int \frac{1}{u} \, du \), which is known to be \( \ln |u| + C \).

Using this, for our problem, when \( u = 1 + t^2 \), the integrated result becomes \( \frac{1}{2} \ln |1 + t^2| + C \). The natural logarithm, denoted by \( \ln \), is a logarithm with a base of \( e \), where \( e \) is roughly equal to 2.718.

• This function is significant in calculus, as it often comes into play while integrating rational or exponential expressions.
• The constant \( C \) is crucial as it accounts for any initial conditions provided.

Once integration is complete, the expression using natural logarithms elegantly presents the solution.

Initial conditions are vital for finding a unique solution to a differential equation. They provide specific values for the constants in the general solution of the differential equation.

In this example, the solution we derived is \( y(t) = \frac{1}{2} \ln |1 + t^2| + C \). The term \( C \) represents the integration constant, which adjusts based on initial conditions.

• If an initial condition is given, such as \( y(t_0) = y_0 \), you plug these values into the general solution.
• This substitution allows the determination of \( C \), making the solution specific to the problem's given situation.
• In problems without specified initial conditions, \( C \) remains arbitrary, and the solution is considered "general."

Therefore, initial conditions personalize the mathematical model to match specific scenarios, giving us a unique solution rather than a family of solutions.