Problem 22 - Differential Equations and Linear Algebra

Question

Determine whether the given function is an integrating factor for the given differential equation. $$I(x)=\sec x,\left[2 x-\left(x^{2}+y^{2}\right) \tan x\right] d x+2 y d y=0$$

Step-by-Step Solution

Answer

The given function \(I(x) = \sec x\) is not an integrating factor for the given differential equation, as the mixed partial derivatives after multiplying the equation by the integrating factor are not equal, i.e., \(\frac{\partial M}{\partial y} \neq \frac{\partial N}{\partial x}\).

Step 1: Multiply the differential equation by the integrating factor

Multiply the given differential equation by the candidate integrating factor, \(I(x) = \sec x\), and then rewrite the equation: \[(2x\sec x - (x^2 +y^2) \sec x \tan x) dx + (2y\sec x) dy = 0\]

Step 2: Check if the resulting equation is exact

To check if the equation is exact, we need to verify if its mixed partial derivatives are equal i.e., if \(\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}\), where \(M(x, y)\) is the coefficient of \(dx\) and \(N(x, y)\) is the coefficient of \(dy\). In this case, we have: \[M(x, y) = 2x\sec x - (x^2 + y^2) \sec x \tan x\] \[N(x, y) = 2y\sec x\] Now, let's compute the partial derivatives: \(\frac{\partial M}{\partial y} = -2y\sec x\tan x\) \(\frac{\partial N}{\partial x} = 2y\sec x\tan x\) Since the mixed partial derivatives are not equal, i.e., \(\frac{\partial M}{\partial y} \neq \frac{\partial N}{\partial x}\), the given integrating factor, \(I(x) = \sec x\), is not an integrating factor for the given differential equation. Thus, we conclude that the given function \(\sec x\) is not an integrating factor for the given differential equation.

Key Concepts

Exact Differential Equations

An exact differential equation is a specific type of equation that can be characterized by a special relationship between its components. To understand this, it is helpful to look at it in the form of \begin{align*}M(x, y)dx + N(x, y)dy = 0ewlineewlineewlineewline\text{where }M \text{ and }N \text{ are functions of }x \text{ and }y. For an equation to be exact, there must exist a function }F(x, y)\text{ such that: }ewlineewlineewlineewline\frac{\partial F}{\partial x} = M \text{ and } \frac{\partial F}{\partial y} = N.\end{align*}This implies a sort of symmetry: the partial derivative of \(M\) with respect to \(y\) must be equal to the partial derivative of \(N\) with respect to \(x\). In other words, \(\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}\). This property allows us to integrate \(M\) and \(N\) to find the function \(F\), which is the solution to our differential equation.

Integrating Factor Method

The integrating factor method is a powerful tool for working with non-exact differential equations. It hinges on finding a function, known as an integrating factor, which when multiplied by the differential equation, renders it exact. The key to this method is in identifying the correct integrating factor, \begin{align*}I(x) \text{ or } I(y),ewlineewlineewlineewline\text{that depends solely on }x \text{ or }y.\end{align*}Once determined, you multiply every term in the differential equation by this integrating factor.

1. This transforms the non-exact equation into one that fits the criteria of an exact differential equation.
2. You then solve the resulting equation using the methods applicable to exact differential equations, potentially finding an implicit solution in terms of \(F(x, y)\).

However, not every function can be an integrating factor. For instance, in our exercise, the integrating factor attempted was \(I(x) = \sec x\), which didn't achieve the goal of making the equation exact, as the condition \(\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}\) was not satisfied after its application.

Partial Derivatives

Partial derivatives are a fundamental concept in multivariable calculus. They describe how a multi-variable function changes with respect to one variable, holding the others constant. When we deal with functions of \(x\) and \(y\), such as \(M(x, y)\) and \(N(x, y)\) in differential equations, it is often necessary to compute these derivatives.By computing \(\frac{\partial M}{\partial y}\) and \(\frac{\partial N}{\partial x}\), we are examining the change in the function \(M\) as \(y\) changes (while keeping \(x\) fixed) and vice versa for \(N\). If the two partial derivatives are equal, they provide insight into the relationship between \(M\) and \(N\), indicating the potential for finding an exact solution. Understanding partial derivatives is essential for analyzing the properties of differential equations and is also crucial when applying methods like the integrating factor method, as they help determine whether the transformed equation is exact.