Problem 19 - Advanced Engineering Mathematics

Question

We have \(d y / d x=y^{\prime} / x^{\prime}=-f(x) / y\) and so, using separation of variables, $$\frac{y^{2}}{2}=-\int_{0}^{x} f(\mu) d \mu+c \quad \text { or } \quad y^{2}+2 F(x)=c$$ . We can conclude that for a given value of \(x\) there are at most two corresponding values of \(y .\) If (0,0) were a stable spiral point there would exist an \(x\) with more than two corresponding values of \(y .\) Note that the condition \(f(0)=0\) is required for (0,0) to be a critical point of the corresponding plane autonomous system \(x^{\prime}=y, y^{\prime}=-f(x)\).

Step-by-Step Solution

Answer

There can be at most two values of \(y\) for a given \(x\). If \((0,0)\) were a stable spiral point, it would lead to more than two \(y\) values, which contradicts the solution.

Step 1: Identify the Given Equation

The exercise provides the equation \(\frac{d y}{d x} = \frac{y^{\prime}}{x^{\prime}} = -\frac{f(x)}{y}\). This equation is meant to be solved using separation of variables.

Step 2: Separate Variables

Rearrange the equation to separate \(y\) and \(x\) terms. We get \(-y \, dy = f(x) \, dx\). Integrating both sides should yield an expression relating \(y\) and \(x\).

Step 3: Integrate Both Sides

Integrate the left side with respect to \(y\) and the right side with respect to \(x\). This gives us the integral equation: \(-\frac{y^2}{2} = -\int_0^x f(\mu) \, d\mu + c\). Simplifying, it becomes \(\frac{y^2}{2} = -\int_0^x f(\mu) \, d\mu + c\).

Step 4: Simplify the Resulting Equation

Rearrange the integrated equation: \(y^2 + 2F(x) = c\), where \(F(x)\) is the antiderivative of \(f(x)\), i.e., \(F(x) = \int_0^x f(\mu) \, d\mu\).

Step 5: Analyze the Solution

The equation \(y^2 + 2F(x) = c\) implies that for a given \(x\), there can be at most two corresponding values of \(y\) because it is a quadratic equation in \(y\).

Step 6: Determine the Nature of the Critical Point

If \((0,0)\) is a stable spiral point, there would exist an \(x\) with more than two corresponding values of \(y\), which contradicts our result since \(y^2 + 2F(x) = c\) can only have at most two values for \(y\). Hence, \(f(0) = 0\) is required for \((0,0)\) to be a critical point.

Key Concepts

Separation of Variables

Separation of variables is a method to solve differential equations, particularly useful when the equation can be arranged such that each variable appears on a different side of the equation. This step often involves rearranging the equation, so the differentials for each variable are on opposite sides.
For instance, in the provided exercise, we begin with the equation \(\frac{d y}{d x} = -\frac{f(x)}{y}\). The goal is to separate the variables, meaning position \(y\) on one side and \(x\) on the other, resulting in \(-y \, dy = f(x) \, dx\).
By isolating the variables, we can then integrate each side individually. This is why separation of variables is one of the essential techniques in solving ordinary differential equations.

Critical Points

Critical points in the context of differential equations refer to the places where the system's state doesn't change over time. It's where the derivative is zero. Identifying critical points helps in understanding the system's stability and behavior.
In the given example, to determine if \((0,0)\) is a critical point, the condition \(f(0) = 0\) must be satisfied. This requirement is crucial because it means the rate of change \(x'\) and \(y'\) is zero at that point, leading to no movement in the autonomous system.
When analyzing the nature of these points, we might determine whether they are stable; often involving deeper criteria beyond just solving the differential equations. In the step-by-step solution, analyzing \((0,0)\) ensures no extra unexpected solutions emerge.

Autonomous Systems

An autonomous system is a set of differential equations in which the variables do not explicitly depend on the independent variable, commonly time. They are common in modeling scenarios where the system's behavior is influenced by the current state and not the sequence of events leading to it.
For the plane autonomous system \(x' = y\) and \(y' = -f(x)\), the variables change according to their current locations, independent of the parameter, usually time.
Such systems naturally fit the context of analyzing critical points, which do not change over time. In our problem, the indication that \((0,0)\) is not a stable spiral point helps conclude that the equation imposes a limitation on overlapping solutions for \(y\).

Integration Techniques

Integration techniques are the methods applied to calculate integrals, which are essential when solving differential equations. After separating variables, the next step is integrating each side of the equation.
The equation \(-y \, dy = f(x) \, dx\) integrates to \(-\frac{y^2}{2} = -\int_0^x f(\mu) \, d\mu + c\). This requires finding an antiderivative \(F(x)\) of \(f(x)\), leading to \(\frac{y^2}{2} = -F(x) + c\).
The proficiency in different integration methods, like substitution or partial fractions, is useful to simplify the integration process. Thus, mastering these techniques is crucial for solving differential equations effectively.