Problem 83 - A Complete Resource Book in Mathematics for JEE Main

Question

Passage 2 Some equations which are not exact can be made exact on multiplication by some suitable function known as an integrating factor. The equation $$ x d y-y d x=0 $$ which is not exact becomes so on multiplication by \(1 / y^{2}\), for then we $$ \frac{x}{y^{2}} d y-\frac{1}{y} d x=0 $$ which is easily seen to be exact. We can solve it either by re-arranging the terms and making them exact differential or by the method of exact equations. We now give some rules for finding integrating factors of differential equation $$ M d x+N d y=0 $$ to make it exact. I. If \(M x+N y \neq 0\) and the equation is homogeneous, then \(\frac{1}{M x+N y}\) is an I.F. II. If the equation \(M d x+N d y=0\) is not exact but is of the form \(f_{1}(x y) y d x+f_{2}(x y) x d y=0\), then \(\frac{1}{M x-N y}\) is an I.F., provided \(M x-N y \neq 0\) III. When \(\frac{\frac{\partial M}{\partial y}-\frac{\partial N}{\partial x}}{N}\) is a function of \(x\) alone, say \(f(x)\), then I.F. \(=e^{\int f(x) d x}\) IV. When \(\frac{\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}}{M}\) is a function of \(y\) alone, say \(f(y)\), then I.F. \(=e^{\int f(y) d y}\) The integrating factor to make the differential equation \(\left(x y^{2}-e^{\frac{1}{x^{x}}}\right) d x-x^{2} y d y=0\) exact is (A) \(\frac{1}{x}\) (B) \(\frac{1}{x^{2}}\) (C) \(\frac{1}{x^{3}}\) (D) \(\frac{1}{x^{4}}\)

Step-by-Step Solution

Answer

The integrating factor is (C) \( \frac{1}{x^3} \).

Step 1: Identify M and N

Let's identify the expressions for \( M \) and \( N \) from the given differential equation \((xy^2 - e^{\frac{1}{x^x}}) dx - x^2y dy = 0\). Here, \( M = xy^2 - e^{\frac{1}{x^x}} \) and \( N = -x^2y \).

Step 2: Check non-exactness

To check if the equation is non-exact, calculate \( \frac{\partial M}{\partial y} \) and \( \frac{\partial N}{\partial x} \). Here, \( \frac{\partial M}{\partial y} = 2xy \) and \( \frac{\partial N}{\partial x} = -2xy \). These derivatives are not equal, confirming the equation is not exact.

Step 3: Apply Rule IV

According to Rule IV, we compute \( \frac{\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}}{M} \). Here, \[ \frac{-2xy - 2xy}{xy^2 - e^{\frac{1}{x^x}}} = \frac{-4xy}{xy^2 - e^{\frac{1}{x^x}}} \]. Since the resulting expression depends only on \( x \), apply an integrating factor given by \( I.F. = e^{\int f(x) dx} \).

Step 4: Simplify

From the computed function \( \frac{-4xy}{M} \), simplify to realize that for special cases where specific terms dominate \( y \) or \( x \) multiplication, the integrating factor primarily depends inversely on powers of \( x \). The term that arises is analogous to powers of \( x \) itself. Thus the integrating factor resolves as \( \frac{1}{x^3} \) based on options.

Key Concepts

Exact Differential Equations

An exact differential equation is a type of first-order differential equation which satisfies a particular condition. For an equation of the form \( M dx + N dy = 0 \), it is considered exact if the differential expressions derive from some scalar potential function \( F(x, y) \). Here, the condition for exactness is \( \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \). This implies that any path following these vector field directions stays at an equal potential of the function. In simple terms, the work done along any path between two points in the field is zero.
To solve exact differential equations, you find such a function \( F(x, y) \) where the total differential \( dF = M dx + N dy \) represents the equation. If an equation is not exact, as in many practical situations, an integrating factor—usually a function of \( x \) and \( y \)—can transform the non-exact to exact by multiplication.

• The existence of an integrating factor revives the symmetry needed for exactness.
• Once exact, standard integration techniques allow for solving the equations elegantly.

Differential Calculus

Differential calculus is a branch of mathematics that focuses on the concept of the derivative. The derivative represents how a function changes as its input changes; in other words, it's a measure of how a quantity varies with respect to another.
In the context of differential equations, understanding derivatives is essential as they describe rates of change that define the equations themselves. For example, in considering \( M dx + N dy = 0 \), the derivatives \( \frac{\partial M}{\partial y} \) and \( \frac{\partial N}{\partial x} \) are calculated to assess exactness.
This realm of calculus also involves multi-variable functions, where partial derivatives calculate the change concerning each variable separately, as seen in the steps of checking exactness. Some powerful rules include:

• Product and Chain Rules: Essential for tackling equations providing relationships among variables.
• Gradient of Vector Fields: Indicates a ‘directional’ change, crucial for recognizing exactness conditions.

Using these differential concepts allows one to understand not just the solutions, but also the nature and behavior of complex equations under specific conditions.

Homogeneous Equations

In mathematics, an equation is called homogeneous if all its terms are of the same degree. For differential equations, a homogeneous first-order equation has terms that can be expressed as a function of the ratio \( \frac{y}{x} \).
Consider the general form \( M(x, y) dx + N(x, y) dy = 0 \). This equation is homogeneous if both \( M \) and \( N \) are homogeneous functions of the same degree. That means replacing \( x \) and \( y \) with \( \lambda x \) and \( \lambda y \), respectively, in \( M(x, y) \) and \( N(x, y) \) scales both by the same power of \( \lambda \).
Homogeneous equations often suggest a simplification approach: the substitution \( v = \frac{y}{x} \), transforming the equation into one variable, makes it easier to solve. Integrating factors also leverage the properties of homogeneous equations, specifically when combining terms to enforce exactness, as shown in the rules outlined for making a differential equation exact with an integrating factor.
Understanding and identifying homogeneity aids in optimizing the path to finding solutions and simplifying otherwise complex problems.