91Ó°ÊÓ

Integration is a critical concept in solving differential equations. It involves finding a function whose derivative is equal to the given function. This process is known as finding the antiderivative or the integral. In our exercise, we started with the second derivative given by \[ f''(\theta) = \sin(\theta) + \cos(\theta). \] By integrating this expression with respect to \( \theta \), we found the first derivative: \[ f'(\theta) = -\cos(\theta) + \sin(\theta) + C, \] where \( C \) is the constant of integration.

• The integration of \( \sin(\theta) \) resulted in \( -\cos(\theta) \).
• The integration of \( \cos(\theta) \) resulted in \( \sin(\theta) \).

Integration allowed us to reverse the differentiation process and move a step closer to finding \( f(\theta) \). It is pivotal to include the constant of integration \( C \), as it accounts for any constant term that may have been differentiated to zero.

When solving differential equations, we often encounter constants of integration. Initial conditions help us determine these constants by providing specific values that the solution must satisfy. For the problem at hand, we used two initial conditions: \( f(0) = 3 \) and \( f'(0) = 4 \). These conditions allowed us to solve for the constants \( C \) and \( D \) in our antiderivatives.
By substituting \( \theta = 0 \) into the first derivative, we established that \[ -1 + 0 + C = 4, \] leading to \( C = 5 \). This computation was essential in determining the correct form of the first derivative.
Subsequently, we used the other initial condition, \( f(0) = 3 \), to find \( D \) in the expression \[ f(\theta) = -\sin(\theta) - \cos(\theta) + 5\theta + D. \] We calculated \[ -1 + D = 3, \] thus finding that \( D = 4 \). The initial conditions anchor the solution to specific values, eliminating ambiguity caused by the constants of integration.

Trigonometric functions like \( \sin(\theta) \) and \( \cos(\theta) \) are often found in differential equations due to their periodic and oscillatory nature. In our exercise, the presence of \( \sin(\theta) \) and \( \cos(\theta) \) in the second derivative signifies an equation involving circular motion or wave patterns.

• When integrating \( \sin(\theta) \), we receive \( -\cos(\theta) \) because the derivative of \( -\cos(\theta) \) is \( \sin(\theta) \).
• Similarly, the integration of \( \cos(\theta) \) results in \( \sin(\theta) \), considering that the derivative of \( \sin(\theta) \) is \( \cos(\theta) \).

These functions' properties greatly aid in solving differential equations, especially in modeling scenarios with periodic behaviors. Recognizing the relationships between derivatives and integrals of these trigonometric functions is fundamental when undertaking problems like our current exercise.