Problem 2 - Calculus 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

(ii) When \(M x-N y \neq 0\) and in the differential equation \(M d x+N d y=0, M=y f_{1}(x, y)\) kand \(N=x f_{2}(x, v)\), then integrating factor of differential equation is \(=\frac{1}{M x-N y}\)

Step-by-Step Solution

Answer

The integrating factor is \(\frac{1}{M x - N y}\) if \(M x - N y \neq 0\).

Step 1: Understand the Problem

We need to find the integrating factor for the given differential equation when \(M x - N y eq 0\). The differential equation given is in the form \(M dx + N dy = 0\) with \(M = y f_1(x, y)\) and \(N = x f_2(x, y)\).

Step 2: Identify Components

Here, \(M = y f_1(x, y)\) and \(N = x f_2(x, y)\) are parts of the differential equation. We are also given the condition \(M x - N y eq 0\). We need to focus on finding when and how an integrating factor can be used.

Step 3: Apply Integrating Factor Formula

The problem suggests that the integrating factor for the differential equation is \(\frac{1}{M x - N y}\). An integrating factor is used to convert a non-exact differential equation into an exact one.

Step 4: Use Given Condition

Ensure the condition \(M x - N y eq 0\) holds true to use the integrating factor \(\frac{1}{M x - N y}\). This assures that the denominator is non-zero, making the integrating factor valid.

Key Concepts

Differential Equations

A differential equation is a mathematical equation that relates some function with its derivatives. In simple terms, it involves rates of change and the functions with respect to which changes occur. It is a common way to express physical phenomena:

• Those involving motion
• Heat
• Electric fields
• Population growth

Understanding differential equations is crucial for modeling various situations in science and engineering. A basic example is Newton's second law, which can be represented as a differential equation. In our context, we are looking at a specific type of differential equation where the form is given by \(M\, dx + N\, dy = 0\). Here, \(M\) and \(N\) are functions of \(x\) and \(y\), and the equation can typically be solved when simplified either by direct integration, or by using special methods such as integrating factors.

Non-exact Equations

Non-exact equations are types of differential equations where the expression \(M\, dx + N\, dy = 0\) does not directly equalize to an exact differential, making it unsolvable by simple integration. To qualify as exact, a differential equation must meet the condition \(\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}\). When this condition isn't met, we refer to the equation as non-exact.
A non-exact equation means there is an openness, or in other terms, a `gap' in how the functions \(M\) and \(N\) interact through their partial derivatives.

• These require extra work via methods such as using an integrating factor.
• Non-exact equations often come up when dealing directly with physical systems that have not yet been simplified or linearized in some way.

Recognizing when a differential equation is non-exact is the first step toward finding the correct integrating factor to solve it.

Exact Equations

An exact equation is a type of differential equation where the expression \(M\, dx + N\, dy = 0\) satisfies the condition \(\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}\), meaning it can be directly integrated to find a solution. When this condition holds true, the functions \(M\) and \(N\) perfectly fit together to create a closed path or loop.
Why is this important? Well, exactness simplifies the solving process significantly.

• An exact equation can often be solved simply by integrating \(M\) with respect to \(x\) and \(N\) with respect to \(y\).
• No further adjustments or external factors are needed to move forward to the solution.

Understanding the distinctions between exact and non-exact equations is a core part of mastering differential equations, as this knowledge dictates the solving strategy.

Mathematical Formulas

Mathematical formulas are essential tools in solving equations, including differential equations. In the context of our problem, the formula for an integrating factor is crucial. It turns a non-exact differential equation into an exact one, making it solvable. The formula we are interested in for the integrating factor here is given by:\[\text{Integrating Factor} = \frac{1}{M x - N y}\]Using this formula effectively requires:

• Confirming that \(M x - N y eq 0\) to ensure the denominator isn't zero, which would make the factor undefined.
• Once the equation is exact, traditional methods of integration can find the solution.

Formulas in mathematics, such as the one for an integrating factor, often provide a structured pathway from problem to solution. Restructuring a differential equation using an integrating factor can make an otherwise complex problem manageable.