Question

For the following exercises, solve the differential equations. \(\frac{d y}{d \theta}=\sin ^{4}(\pi \theta)\)

Short answer

The solution is \( y = -\frac{1}{4} (3\theta + \frac{1}{\pi}\sin(2\pi \theta) + \frac{1}{2}\sin(4\pi \theta)) + C \).

Step-by-step solution

Identify the type of differential equation

The given differential equation is \( \frac{d y}{d \theta} = \sin^4(\pi \theta) \). This is a first-order, separable ordinary differential equation.

Separate variables

Rearrange the equation to separate the variables: \( dy = \sin^4(\pi \theta) \, d\theta \). This allows us to integrate both sides separately with respect to their own variables.

Integrate both sides

Integrate both sides of the equation: \( \int dy = \int \sin^4(\pi \theta) \, d\theta \). This results in \( y = -\frac{1}{4} \left(3\theta + \frac{1}{\pi}\sin(2\pi \theta) + \frac{1}{2}\sin(4\pi \theta)\right) + C \), where \( C \) is the integration constant.

Simplify the equation

The solution becomes \( y = -\frac{1}{4} \left(3\theta + \frac{1}{\pi}\sin(2\pi \theta) + \frac{1}{2}\sin(4\pi \theta)\right) + C \). The integration is complete, and the expression is simplified. This equation represents the family of solutions dependent on the constant \( C \).

Key concepts

Separable Ordinary Differential Equations

A separable ordinary differential equation is a special type of differential equation where the variables can be separated to different sides of the equation. This means that one variable and its differential can be placed on one side of the equation, while the other variable and its differential are placed on the other. This separation allows for simpler integration and solution.In the exercise provided, the equation given is \( \frac{d y}{d \theta} = \sin^4(\pi \theta) \). It is recognized as a separable ordinary differential equation because we can rewrite it as:

• \( dy = \sin^4(\pi \theta) \, d\theta \)

This rearrangement easily lets us integrate each side separately. Because the separation of variables is possible, solving the equation becomes more straightforward. We integrate \( dy \) with respect to \( y \) and \( \sin^4(\pi \theta) \) with respect to \( \theta \).

First-Order Differential Equations

A first-order differential equation involves the first derivative of the function and is often written in the form \( \frac{dy}{dx} = f(x) \) or similar expressions. These equations are fundamental in calculus and they describe how a quantity changes with respect to another variable.For the exercise equation, \( \frac{d y}{d \theta} = \sin^4(\pi \theta) \), we identify this as a first-order differential equation because it involves just the first derivative \( \frac{d y}{d \theta} \). Solving such an equation requires integration, since we need to find the original function \( y \) whose rate of change is given by the equation. First-order equations often form the basis for modeling real-world scenarios where rates and changes are essential components of the analysis.

Integration Constant in Solutions

When integrating to solve a differential equation, an important element is the integration constant often denoted as \( C \). This constant represents an entire family of solutions rather than just a single solution.In the process of integrating both sides of the given separable equation:

• \( \int dy = \int \sin^4(\pi \theta) \, d\theta \)

We obtain the solution:

• \( y = -\frac{1}{4} \left(3\theta + \frac{1}{\pi}\sin(2\pi \theta) + \frac{1}{2}\sin(4\pi \theta)\right) + C \)

The constant \( C \) plays a crucial role, as it allows the function \( y \) to take on many forms depending on its value. This means the solution isn't just represented by a single curve but by an infinite set of curves on the graph, all parallel and differing by their \( C \) values. This is why in practical applications, additional information or conditions are required to determine the specific value of \( C \).