Problem 17 - Advanced Engineering Mathematics

Question

Determine whether the given differential equation is exact. If it is exact, solve it. $$ (\tan x-\sin x \sin y) d x+\cos x \cos y d y=0 $$

Step-by-Step Solution

Answer

The equation is exact, with solution \(-\ln |\cos x| + \sin y \cos x = C\).

Step 1: Understanding Exact Differential Equations

An exact differential equation has the form \(M \, dx + N \, dy = 0\), where \(\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}\). In the problem, \(M(x, y) = \tan x - \sin x \sin y\) and \(N(x, y) = \cos x \cos y\).

Step 1: Calculate Partial Derivatives

First, compute \(\frac{\partial M}{\partial y}\) and \(\frac{\partial N}{\partial x}\). For \(M = \tan x - \sin x \sin y\), the derivative with respect to \(y\) is \(-\sin x \cos y\). For \(N = \cos x \cos y\), the derivative with respect to \(x\) is \(-\sin x \cos y\).

Step 2: Compare Partial Derivatives

To determine if the equation is exact, compare \(\frac{\partial M}{\partial y}\) with \(\frac{\partial N}{\partial x}\). Both are equal: \(\frac{\partial M}{\partial y} = -\sin x \cos y = \frac{\partial N}{\partial x}\). Thus, the differential equation is exact.

Step 3: Solve the Exact Differential Equation

To solve the exact differential equation, find a function \(F(x, y)\) such that \(\frac{\partial F}{\partial x} = M\) and \(\frac{\partial F}{\partial y} = N\). First, integrate \(M\) with respect to \(x\).

Step 4: Integration with respect to x

Integrate \(M = \tan x - \sin x \sin y\) with respect to \(x\) to get \(F(x, y) = -\ln |\cos x| + \sin y \cos x + g(y)\), where \(g(y)\) is an arbitrary function of \(y\).

Step 5: Determine g(y)

Differentiate \(F(x, y) = -\ln |\cos x| + \sin y \cos x + g(y)\) with respect to \(y\) to find \(g'(y)\). Set this equal to \(N(x, y) = \cos x \cos y\): \( \cos x \cos y + g'(y) = \cos x \cos y\). This gives \(g'(y) = 0\), implying \(g(y) = C\), a constant.

Step 6: Formulate the Solution

The solution to the differential equation is found by setting \(F(x, y) = \text{constant}\). Thus, the solution is \(-\ln |\cos x| + \sin y \cos x = \text{constant}\).

Key Concepts

Partial Derivatives

In calculus, partial derivatives help us understand how a function changes as one of its variables changes, while the others are kept constant.
When dealing with exact differential equations, partial derivatives guide us in verifying the exactness of the equation.In the given differential equation:

• We have two expressions:
  • \(M(x, y) = \tan x - \sin x \sin y\)
  • \(N(x, y) = \cos x \cos y\)
• The idea is to compute the partial derivative of \(M\) with respect to \(y\), and \(N\) with respect to \(x\).

If the partial derivatives are equal, it confirms the equation's exactness.
In this example:

• \(\frac{\partial M}{\partial y} = -\sin x \cos y\)
• \(\frac{\partial N}{\partial x} = -\sin x \cos y\)

Both derivatives being the same implies that the given equation is exact and ready for solving.

Integration Techniques

Once we know the differential equation is exact, our next step is using integration to find a potential function \(F(x, y)\) that describes the solution.
Solving an exact differential equation involves integrating one of the components of our equation.In this case:

• We start by integrating \(M(x, y)\) with respect to \(x\).

Upon integrating \(M = \tan x - \sin x \sin y\):

• The result is \(F(x, y) = -\ln |\cos x| + \sin y \cos x + g(y)\).

Here, \(g(y)\) is an arbitrary function that appears because of the integration.
We must solve for this function, which will bring us to the next step. Integration in exact equations is simple but requires careful management of terms and arbitrary functions.

Differential Equation Solution Steps

With partial derivatives and integration in mind, solving an exact differential equation is a series of logical steps:

• Verify Exactness: Confirm the equation's exactness through partial derivatives.
• Integrate with Respect to x: Perform integration on \(M(x, y)\) to find an expression for the potential function \(F(x, y)\).
• Include g(y): The integration step provides a function \(g(y)\), an integral constant with respect to \(x\).
• Differentiate and Solve: Differentiate \(F(x, y)\) with respect to \(y\) to find \(g'(y)\) and equate it to \(N(x, y)\).
• Solve for g(y): Determine \(g(y)\) by solving \(g'(y) = 0\), which results in \(g(y) = C\), a constant.
• Formulate the Solution: Set \(F(x, y)\) equal to a constant to express the general solution.

Hence, for this exercise, the solution becomes \(-\ln |\cos x| + \sin y \cos x = \text{constant}\).
These steps outline a reliable methodical approach, ensuring clarity in solving exact differential equations.