91Ó°ÊÓ

First-order linear differential equations, like the one in our example, often require an integrating factor to solve them. The integrating factor is a clever device that helps transform a hard-to-solve equation into something much easier to handle.

Consider a differential equation in the form \( y' + P(t)y = Q(t) \). The integrating factor \( \mu(t) \) is found using the formula \( \mu(t) = e^{\int P(t) \, dt} \).

In our case, with \( P(t) = \sec t \), we compute \( \int \sec t \, dt \). The integral of secant is a classic problem in calculus, often resulting in \( \ln |\sec t + \tan t| \).

The integrating factor we derive, \( \mu(t) = e^{\ln |\sec t + \tan t|} = |\sec t + \tan t| \), simplifies beautifully because \( \sec t + \tan t \) is positive between \(-\pi/2 < t < \pi/2 \). We conclude that \( \mu(t) = \sec t + \tan t \), thus easing our journey towards solving the equation.

The general solution of a differential equation provides us with a broad view of all possible solutions. It represents a family of functions based on the constant of integration, often denoted by \( C \).

For the equation given, after applying the integrating factor, we end up with the transformed equation. Once simplified, the equation becomes \( \frac{d}{dt}[y(\sec t + \tan t)] = (\sec t + \tan t) \tan t \).

Both sides are integrated, leading us to \( y(\sec t + \tan t) = \sec t + \tan t - t + C \).

Finally, we solve for \( y \) through division, producing the general solution:

• \( y = 1 - \frac{t}{\sec t + \tan t} + \frac{C}{\sec t + \tan t} \)

This solution encompasses all possible functions that satisfy the differential equation, parametrized by \( C \), and is valid for the interval \(-\pi/2 < t < \pi/2 \).

Trigonometric integrals appear frequently in solving differential equations, especially when dealing with functions like secant and tangent.

In this problem, the trigonometric integral involving \( \sec t \tan t \) and \( \tan^2 t \) are front-and-center. Each of these integrals has standard results:

• \( \int \sec t \tan t \, dt = \sec t + C \)
• \( \int \tan^2 t \, dt \) can be found using the identity \( \tan^2 t = \sec^2 t - 1 \). Upon integration, this gives \( \tan t - t + C \).

The key to managing these integrals is knowing the trigonometric identities and the standard integrals of trigonometric functions. They not only simplify the calculations but also clarify how each component relates to the equation's solution.

Understanding these standard forms and their derivations provides a strong foundation for solving complex differential equations involving trigonometric expressions.