11–18. Solving initial value problems Use the method of your choice to find the solution of the following initial value problems.
y′(x) = 4x csc y, y(0) = π/2

Verified step by step guidance

1. Identify the type of differential equation given: \( y'(x) = 4x \csc y \). Notice that it is separable because the right-hand side can be expressed as a product of a function of \( x \) and a function of \( y \).
2. Rewrite the differential equation in separable form by expressing \( y' = \frac{dy}{dx} \) and rearranging terms to isolate \( y \) and \( x \) on opposite sides: \( \sin y \, dy = 4x \, dx \).
3. Integrate both sides: compute \( \int \sin y \, dy \) on the left and \( \int 4x \, dx \) on the right. Remember to include the constant of integration after integrating.
4. Apply the initial condition \( y(0) = \frac{\pi}{2} \) to solve for the constant of integration. Substitute \( x = 0 \) and \( y = \frac{\pi}{2} \) into the integrated equation.
5. Rewrite the implicit solution to express \( y \) explicitly in terms of \( x \) if possible, or leave it in implicit form as the solution to the initial value problem.

Key Concepts

Separable Differential Equations

A separable differential equation can be written as a product of a function of x and a function of y, allowing variables to be separated on opposite sides of the equation. This enables integration with respect to each variable independently to find the general solution.

Initial Value Problems (IVP)

An initial value problem specifies a differential equation along with a condition that the solution must satisfy at a particular point. This condition helps determine the unique solution by solving for the constant of integration after finding the general solution.

Integration of Trigonometric Functions

Solving the given differential equation involves integrating expressions containing trigonometric functions like cosecant (csc y). Understanding how to integrate these functions, often using identities or substitution, is essential to find the explicit solution.