Problem 76 - University Calculus: Early Transcendentals

Question

Solve the initial value problems $$\frac{d r}{d \theta}=3 \cos ^{2}\left(\frac{\pi}{4}-\theta\right), \quad r(0)=\frac{\pi}{8}$$

Step-by-Step Solution

Answer

The solution is \( r(\theta) = \frac{3}{2} \theta + \frac{3}{4} \sin\left(\frac{\pi}{2} - 2\theta\right) + \frac{\pi - 6}{8} \).

Step 1: Understand the Problem

We are given a differential equation \( \frac{dr}{d\theta} = 3 \cos^2\left(\frac{\pi}{4} - \theta\right) \) with the initial condition \( r(0) = \frac{\pi}{8} \). Our task is to find the function \( r(\theta) \) that satisfies both the differential equation and the initial condition.

Step 2: Set Up the Integral

To solve \( \frac{dr}{d\theta}=3 \cos^{2}\left(\frac{\pi}{4}-\theta\right) \), we need to integrate both sides with respect to \( \theta \). We set up the integral as follows: \[ r(\theta) = \int 3 \cos^{2}\left(\frac{\pi}{4} - \theta\right) \, d\theta + C \] where \( C \) is the constant of integration.

Step 3: Determine the Integral

To solve \( \int 3 \cos^2\left(\frac{\pi}{4} - \theta\right) \, d\theta \), we use the identity \( \cos^2(x) = \frac{1 + \cos(2x)}{2} \). This gives us:\[ \int 3 \cos^{2}\left(\frac{\pi}{4} - \theta\right) \, d\theta = \int \frac{3}{2} (1 + \cos\left(\frac{\pi}{2} - 2\theta\right)) \, d\theta \]Simplifying the integration leads to:\[ \frac{3}{2} \left( \theta + \frac{1}{2} \sin\left(\frac{\pi}{2} - 2\theta\right) \right) + C \]So, \( r(\theta) = \frac{3}{2} \theta + \frac{3}{4} \sin\left(\frac{\pi}{2} - 2\theta\right) + C \).

Step 4: Apply Initial Condition

We apply the initial condition \( r(0) = \frac{\pi}{8} \) to find the constant \( C \):\[ \frac{3}{2} \cdot 0 + \frac{3}{4} \sin\left(\frac{\pi}{2} - 0\right) + C = \frac{\pi}{8} \]Since \( \sin\left(\frac{\pi}{2}\right) = 1 \), this simplifies to:\[ \frac{3}{4} + C = \frac{\pi}{8} \]Solving for \( C \) gives us: \( C = \frac{\pi}{8} - \frac{3}{4} = \frac{\pi - 6}{8} \).

Step 5: Write the Solution

Substitute the value of \( C \) back into the expression for \( r(\theta) \):\[ r(\theta) = \frac{3}{2} \theta + \frac{3}{4} \sin\left(\frac{\pi}{2} - 2\theta\right) + \frac{\pi - 6}{8} \]This is the solution that satisfies both the differential equation and the initial condition.

Key Concepts

Differential Equations

Differential equations describe the relationship between a function and its derivatives. These equations are essential in modeling various real-world phenomena, such as motion, heat, and waves.
In the above problem, we are dealing with a first-order differential equation. This type involves derivatives of the function itself, not higher-order derivatives, and is represented as:

• \( \frac{dr}{d\theta} = 3 \cos^2 \left( \frac{\pi}{4} - \theta \right) \).

Differential equations can be classified as either ordinary (ODEs) or partial (PDEs). ODEs involve functions and derivatives with respect to one independent variable, whereas PDEs involve multiple variables.

• Since \( \frac{dr}{d\theta} \) involves differentiation with respect to \( \theta \), it is an ODE.

To solve these equations, we often need to integrate. This leads us to find a function that satisfies the equation and any given initial conditions, indicating the specifics needed for a unique solution.

Integration Techniques

Integration is a fundamental concept in solving differential equations. It helps us find the original function from its derivative(s). Often, it requires special techniques to simplify and solve equations.

• In the equation, \( \int 3 \cos^2 \left( \frac{\pi}{4} - \theta \right) \, d\theta \),using integration is key to finding \( r(\theta) \).

One useful technique employed here is using a trigonometric identity. Substituting \( \cos^2(x) = \frac{1 + \cos(2x)}{2} \) helps to simplify the integrand.The substitution turns a difficult integral into a more manageable form, allowing us to solve it by:

• Separating it into simpler integrals: \( \int \frac{3}{2} (1 + \cos(\frac{\pi}{2} - 2\theta)) \, d\theta \).

By integrating each component separately, the solution for \( r(\theta) \) can be determined, giving the function needed in line with the differential equation given.

Trigonometric Identities

Trigonometric identities are valuable tools for simplifying integration problems, especially those involving trigonometric functions. They can transform complex expressions into simpler, more solvable forms.
In this exercise, we used the identity:

• \( \cos^2(x) = \frac{1 + \cos(2x)}{2} \).

This identity is essential for integrating square trigonometric functions, which would otherwise be more challenging to handle.Trigonometric identities often provide ways to express complex trigonometric functions in terms of simpler linear functions.

• After using substitution, complex integrands reduce, allowing us to integrate a sum of terms, where each term is far simpler.

Incorporating trigonometric identities into solving differential equations not only simplifies calculations but ensures the correct application of initial conditions. This leads to finding a tailored solution that satisfies all given constraints.