Problem 43 - Advanced Engineering Mathematics

Question

Separating variables we have \(d y /\left(\sqrt{1+y^{2}} \sin ^{2} y\right)=d x\) which is not readily integrated (even by a CAS). We note that \(d y / d x \geq 0\) for all values of \(x\) and \(y\) and that \(d y / d x=0\) when \(y=0\) and \(y=\pi,\) which are equilibrium solutions.

Step-by-Step Solution

Answer

The equilibrium solutions are \( y=0 \) and \( y=\pi \).

Step 1: Identifying the Differential Equation

We start with the given differential equation: \( \frac{dy}{dx} = \sqrt{1+y^2} \sin^2 y \). We need to analyze and understand the behavior of this equation.

Step 2: Analyzing the Sign of the Derivative

We observe that \( \frac{dy}{dx} \geq 0 \) for all values of \( x \) and \( y \). This implies that the function \( y(x) \) is non-decreasing.

Step 3: Finding Equilibrium Solutions

The equation \( \frac{dy}{dx} = 0 \) gives equilibrium solutions. Solving \( \sqrt{1+y^2} \sin^2 y = 0 \), we find the equilibrium points by setting \( \sin^2 y = 0 \), which means \( y = 0 \) and \( y = \pi \), due to \( \sin y = 0 \) at these points.

Step 4: Conclusion About the Behavior of Solutions

Since \( \frac{dy}{dx} = 0 \) at \( y = 0 \) and \( y = \pi \), these are constant solutions (equilibrium) where \( y \) does not change. Given \( \frac{dy}{dx} \geq 0 \) everywhere, any other solution will trend towards these equilibrium values without decreasing.

Key Concepts

Separation of Variables

The method of separation of variables is a powerful tool used to solve certain types of differential equations. It is especially useful when we can algebraically rearrange a differential equation into a particular form. This form allows us to isolate different variables on opposite sides of the equation.
Here's how it works:

• Identify if the differential equation can be rewritten such that all instances of one variable are on one side, and the other variable is on the opposite side.

For example, if we start with a differential equation of the form \( \frac{dy}{dx} = f(y)g(x) \), this can be separated to \( \frac{1}{f(y)}dy = g(x)dx \) by multiplying both sides appropriately to isolate \( dy \) and \( dx \).

Once the equation is separated, each side can be integrated independently. However, not all equations can be separated easily or solved in this manner. In the given example, the differential equation \( \frac{dy}{dx} = \sqrt{1+y^2} \sin^2 y \) leads to a complex integrand which isn't easily solvable by basic integration techniques or computer algebra systems (CAS).
This realization indicates either a need for an alternate approach, or in some cases, finding qualitative properties of the solutions without explicit solving.

Equilibrium Solutions

Equilibrium solutions in differential equations are particularly interesting. They are points where the function remains constant, which means there's no change over time or the independent variable.
To find equilibrium solutions, we set the derivative to zero, \( \frac{dy}{dx} = 0 \). Solving this equation gives us specific values of \( y \) where the function doesn't change.

• For the given equation \( \sin^2 y = 0 \), equilibrium occurs when \( \sin y = 0 \).
  This gives the specific values \( y = 0 \) and \( y = \pi \).
  These values signify points where the change (or derivative) is zero, \( \frac{dy}{dx} = 0 \), meaning the function just sits there without increasing or decreasing.

Equilibria are particularly useful because they can provide insight into the long-term behavior of solutions. Even when explicit solutions are hard to find, knowing equilibrium points can tell us where solutions are likely to stabilize or trend toward over time.

Non-decreasing Functions

A non-decreasing function is one where the output (or \( y \) value in terms of \( x \)) does not decrease. In mathematical terms, if \( \frac{dy}{dx} \geq 0 \) for all points in the domain, the function is non-decreasing.
This concept is straightforward yet telling. In the example, \( \frac{dy}{dx} = \sqrt{1+y^2} \sin^2 y \) is always non-negative given that both \( \sqrt{1+y^2} \) and \( \sin^2 y \) are non-negative for all real \( y \).

Understanding non-decreasing functions is essential in ensuring that any changes to the input result in the same or increased outputs. When applying this concept:

• If a function passes through an equilibrium solution, it doesn't increase or decrease—it stays constant.
• Anywhere else, it must either increase or stay the same, but never dip below a previously attained point.

This gives us the understanding that the solutions of the differential equation will trend toward equilibrium points, while being restricted to non-decreasing behavior, offering insights into the nature and potential behavior of the dynamic system being studied.