Problem 129 - University Calculus: Early Transcendentals

Question

Use a CAS to solve the initial value problems in Plot the solution curves.. $$y^{\prime}=\cos ^{2} x+\sin x, \quad y(\pi)=1$$

Step-by-Step Solution

Answer

The solution to the initial value problem is \(y(x) = \frac{1}{2}x + \frac{1}{4}\sin(2x) - \cos x - \frac{1}{2}\pi\).

Step 1: Understanding the Problem

The given initial value problem involves finding the solution to the differential equation \(y^{\prime}=\cos^2 x + \sin x\) with the initial condition \(y(\pi) = 1\). We want to find the function \(y(x)\) that satisfies this equation and initial condition.

Step 2: Integrating the Right Side

To solve the differential equation \(y^{\prime} = \cos^2 x + \sin x\), integrate the right side with respect to \(x\). The integral involves two parts: the integral of \(\cos^2 x\) and the integral of \(\sin x\).

Step 3: Solving \(\int \cos^2 x \, dx\)

Use the identity \(\cos^2 x = \frac{1 + \cos(2x)}{2}\) to rewrite the integral: \(\int \cos^2 x \, dx = \int \frac{1 + \cos(2x)}{2} \, dx\). This simplifies to \(\frac{1}{2}(x + \frac{1}{2} \sin(2x)) + C_1\).

Step 4: Solving \(\int \sin x \, dx\)

The integral of \(\sin x\) is straightforward: \(\int \sin x \, dx = -\cos x + C_2\). Combine the terms with the other integral to form the full integral solution.

Step 5: Combine and Simplify the Solution

Combine both integrals: \(y(x) = \frac{1}{2}x + \frac{1}{4}\sin(2x) - \cos x + C\). This function represents a general solution before applying the initial condition.

Step 6: Apply the Initial Condition

Use the initial condition \(y(\pi) = 1\) to find the constant \(C\). Substitute \(x = \pi\) into the solution: \(1 = \frac{1}{2}\pi + \frac{1}{4}\sin(2\pi) - \cos \pi + C\). Since \(\sin(2\pi) = 0\) and \(\cos(\pi) = -1\), we solve for \(C\) as follows:\[1 = \frac{1}{2}\pi + 1 + C\]Thus, \(C = 1 - \frac{1}{2}\pi - 1 = -\frac{1}{2}\pi\).

Step 7: Final Solution

Substitute \(C\) back into the equation to get the final solution for the initial value problem:\[y(x) = \frac{1}{2}x + \frac{1}{4}\sin(2x) - \cos x - \frac{1}{2}\pi\].

Key Concepts

Differential Equations

Differential equations play a crucial role in modeling various real-world phenomena. A differential equation, like the one we have explored, involves an unknown function and its derivatives. In this specific problem, we encountered a first-order differential equation: \[ y' = \cos^2 x + \sin x \]Here, \(y\) is the function of \(x\) that we aim to determine. By solving the equation, we find a function that describes the relationship between these variables across different contexts or conditions.

• A solution to a differential equation gives insight into how a system behaves.
• The order & linearity of the equation describe its complexity.
• Initial conditions are often required to find a particular solution.

Differential equations are foundational in science and engineering, facilitating the understanding of various processes involving changes over time or space.

Integration

Integration is a fundamental mathematical process used to find functions that describe accumulated quantities. In solving our differential equation, we need to integrate the right-hand side of the equation:\[ y' = \cos^2 x + \sin x \]
The integration process involves calculating the indefinite integral of both terms on the right side. - **For \(\cos^2 x\)**, it requires a trigonometric identity to simplify it before integrating, resulting in: - \(\int \cos^2 x \, dx = \frac{1}{2}(x + \frac{1}{2} \sin(2x)) + C_1\)- **For \(\sin x\)**, its integration is more straightforward: - \(\int \sin x \, dx = -\cos x + C_2\)These integrals, when combined, form the general solution. Integration transforms differential equations into solvable expressions and is essential for finding the function that fulfills these relations.

Trigonometric Identities

Trigonometric identities transform complex trigonometric expressions into simpler ones, aiding in integration and solving differential equations. In our problem, the identity applied was:\[ \cos^2 x = \frac{1 + \cos(2x)}{2} \]
This identity enabled us to rewrite the integral \( \int \cos^2 x \, dx \) into a more manageable form. It leads to:\[ \int \frac{1 + \cos(2x)}{2} \, dx = \frac{1}{2}(x + \frac{1}{2} \sin(2x)) + C_1 \]
Trigonometric identities are vital tools, especially:

• For simplifying expressions before integration.
• In solving integrals involving trigonometric functions.
• They reveal relationships between trigonometric functions, making calculations feasible.

By mastering these identities, complex mathematical problems become more approachable.

Initial Conditions

Initial conditions are specific values assigned to a function at a particular point, enabling the determination of a particular solution to a differential equation. In our problem, the initial condition was:\[ y(\pi) = 1 \]
This condition allows us to find the constant \(C\) in our solution derived from integration.
Steps include:

• Substituting the initial point into the general solution.
• Solving for the constant \(C\) using known trigonometric values.
• Applying these values grants a unique solution that adheres to the initial constraints.

Initial conditions are critical because they tailor the solution of a differential equation to fit specific scenarios, reflecting the uniqueness of real-world situations.