Problem 67 - Calculus Single Variable

Question

Suppose that \(\phi\) is a function of one variable. The differential equation \(d y / d x=\phi(y / x)\) is said to be homogeneous of degree \(0 .\) Let \(w(x)=y(x) / x .\) Differentiate both sides of the equation \(y(x)=x \cdot w(x)\) with respect to \(x .\) By equating the resulting expression for \(d y / d x\) with \(\phi(y / x),\) show that \(w(x)\) is the solution of a separable differential equation. Illustrate this theory by solving the differential equation \(d y / d x=2 x y /\left(x^{2}+y^{2}\right) .\) For this example, \(\phi(w)=2 w /\) \(\left(1+w^{2}\right)\)

Step-by-Step Solution

Answer

Differentiate and rearrange to find a separable differential equation: \(\frac{dw}{\phi(w) - w} = \frac{dx}{x}\).

Step 1: Express y in terms of w and x

Given that \(w(x) = \frac{y(x)}{x}\), we can express \(y\) as \(y = x \cdot w(x)\), where \(w(x) = \frac{y}{x}\). This equation will be used to compute the derivative of \(y\) with respect to \(x\).

Step 2: Differentiate y with respect to x

Differentiate both sides of the equation \(y(x) = x \cdot w(x)\) with respect to \(x\) using the product rule. This gives \[ \frac{dy}{dx} = w(x) + x \cdot \frac{dw}{dx}. \]

Step 3: Equate the derivative to φ(y/x)

Set the expression for \(\frac{dy}{dx}\) equal to \(\phi\left(\frac{y}{x}\right)\). Since \(\frac{y}{x} = w(x)\), this implies \(\phi\left(\frac{y}{x}\right) = \phi(w)\). Therefore, equate \(w(x) + x \cdot \frac{dw}{dx} = \phi(w)\).

Step 4: Show the separable differential equation

Rearrange the equation \(w(x) + x \cdot \frac{dw}{dx} = \phi(w)\) to extract the variable \(w\) on one side. This gives \[ x \cdot \frac{dw}{dx} = \phi(w) - w(x). \] Rewrite this as \[ \frac{dw}{\phi(w) - w} = \frac{dx}{x}, \] which is a separable differential equation.

Step 5: Substitute φ(w) in the example equation

For the given example, \(\phi(w) = \frac{2w}{1+w^2}\). Substitute \(\phi(w) = \frac{2w}{1+w^2}\) into the separable equation \[ \frac{dw}{\frac{2w}{1+w^2} - w} = \frac{dx}{x}. \]

Step 6: Simplify and solve the differential equation

Simplify the equation: \[ \frac{dw}{\frac{2w - w(1+w^2)}{1+w^2}} = \frac{dx}{x}. \] This further simplifies to \[ \frac{dw}{\frac{w(1-w^2)}{1+w^2}} = \frac{dx}{x}. \] Simplifying the fraction, integrate both sides: \[ \int \frac{(1+w^2)}{w(1-w^2)} \, dw = \int \frac{dx}{x}. \] The solution involves solving these integrals.

Key Concepts

Understanding Separable Differential Equations

A separable differential equation is a type of differential equation in which the variables can be separated on each side of the equation. This makes them easier to solve by integration. In mathematical terms, an equation is considered separable if it can be written in the form:

• \( \frac{dy}{dx} = g(y)h(x) \)

Here, the expression \( g(y) \) contains only the variable \( y \), and \( h(x) \) contains only \( x \).
To solve it, we rearrange the equation such that all terms involving \( y \) are on one side, and all terms involving \( x \) are on the other. In this exercise, starting from \( \frac{dy}{dx} = \phi(y/x) \), through differentiation and substitution, we derived a form that allowed separation of variables and transformed it into a separable equation.
The ability to separate variables allows us to solve the equation by integrating each side independently. This often results in finding a function \( y = f(x) \) that satisfies the differential equation.

Using the Product Rule in Differentiation

The product rule is a fundamental technique in calculus used to differentiate expressions where two functions are multiplied together. When you have a product of two functions, say \( u(x) \) and \( v(x) \), the derivative of their product is determined by:

• \( \frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x) \)

In simple terms, this means you take the derivative of the first function and multiply it by the second, and then do the reverse and add the results.
In the original exercise, when finding \( \frac{dy}{dx} \) for \( y = xw(x) \), the product rule helped to expand the expression into \( w(x) + x \cdot \frac{dw}{dx} \). Without the product rule, differentiating a product of functions would be cumbersome and error-prone.
Understanding when and how to use the product rule is crucial for successfully differentiating complex expressions in differential equations.

The Role of Integration in Solving Differential Equations

Integration plays a vital role in solving differential equations as it helps to find the function that satisfies the given equation. After separating the variables in a differential equation, the next step is often to integrate both sides.
For instance, if the differential equation is \( \frac{dy}{dx} = g(y)h(x) \), then by separating the variables, it becomes \( \int \frac{1}{g(y)} \, dy = \int h(x) \, dx \).

• Each side is integrated with respect to its variable.
• The integral may result in a family of functions, typically represented with a constant of integration \( + C \).

In the provided example, once the equation was simplified to a form involving separable variables, each side was integrated to solve for \( w(x) \) and ultimately \( y(x) \).
Integration enables us to transition from the rate of change relationship determined by the differential equation to a specific function describing the behavior of the system.

Deep Dive into Differential Equations

Differential equations are equations that involve an unknown function and its derivatives. They are a key mathematical tool used to solve problems involving rates of change in fields like physics, engineering, and economics.

• They express the relationship between a function and its derivatives.
• The unknown function is typically denoted as \( y(x) \), where \( y \) is dependent on \( x \).

Differential equations can be classified based on various factors, such as order and linearity. In this exercise, the equation \( \frac{dy}{dx} = \phi(y/x) \) was described as a homogeneous differential equation of degree zero.
Such equations often transform into simpler forms, allowing the use of methods like separation of variables. Understanding differential equations and their solutions helps us describe a wide array of natural phenomena and technologies with mathematical precision. By practicing these methods, one becomes adept at transforming and solving these meaningful mathematical constructs.