Problem 81 - 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 solution of the differential equation \(\left(x^{2} y-2 x y^{2}\right) d x-\left(x^{3}-3 x^{2} y\right) d y=0\) is (A) \(\frac{x}{y}-2 \log x+3 \log y=c\) (B) \(\frac{x}{y}+2 \log x+3 \log y=c\) (C) \(\frac{x}{y}-2 \log x-3 \log y=c\) (D) None of these

Step-by-Step Solution

Answer

The correct solution is (A) \(\frac{x}{y}-2 \log x+3 \log y=c\).

Step 1: Identify the Problem

We have a differential equation given by \((x^2 y - 2xy^2) dx - (x^3 - 3x^2 y) dy = 0\). We need to determine if it is exact and solve it to find the correct option among the given choices.

Step 2: Calculate Partial Derivatives

To check the exactness of the differential equation, calculate the partial derivative of \(M = x^2 y - 2x y^2\) with respect to \(y\), and \(N = -(x^3 - 3x^2 y)\) with respect to \(x\). \[\frac{\partial M}{\partial y} = x^2 - 4xy\]\[\frac{\partial N}{\partial x} = -3x^2 + 6xy\]

Step 3: Determine Exactness

The differential equation is exact if \(\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}\). Calculating these gives:\[\frac{\partial M}{\partial y} = x^2 - 4xy\]\[\frac{\partial N}{\partial x} = -3x^2 + 6xy\]Since they are not equal, the equation is not exact.

Step 4: Find Integrating Factor Using Homogeneity

Check the conditions for homogeneity. The differential equation is homogeneous as both expressions \(x^2 y - 2xy^2\) and \( -(x^3 - 3x^2 y)\) are of the same degree, 3. Apply the rule for homogeneous equations, using \(\frac{1}{Mx - Ny}\) as the integrating factor, where \(M x + N y eq 0\).

Step 5: Find the Correct Equation

After determining the integrating factor (IF) and multiplying the entire equation, we integrate it to get to the solution. The new form from integrating factors helps us derive similarity with one of the given options. By integrating and simplifying, we arrive at an equation of the form \(\frac{x}{y} - 2 \log x + 3 \log y = c\).

Step 6: Match with Options

Compare the derived solution with the options provided. The derived solution \(\frac{x}{y} - 2 \log x + 3 \log y = c\) matches option (A). Therefore, option (A) is the correct answer.

Key Concepts

Integrating Factor

An integrating factor is a function used to multiply a given non-exact differential equation to convert it into an exact one. This makes the equation solvable by standard methods for exact equations. Understanding the integrating factor is essential for solving differential equations that don't naturally fit into simpler solution methods.

• For homogeneous differential equations, where each term is of the same degree, the integrating factor can often be found using the expression \( \frac{1}{Mx + Ny} \).
• In some cases, partial derivatives are used to determine if a function of only \(x\) or \(y\) will work as an integrating factor, leading to forms like \( e^{\int f(x) \, dx} \) or \( e^{\int f(y) \, dy} \).

The primary goal of finding an integrating factor is to simplify the given differential equation so you can integrate it easily. Once the equation is exact, it can be solved by integrating with respect to one variable while treating others as constants.

Homogeneous Differential Equation

A homogeneous differential equation is one where all terms can be expressed in terms of variables raised to the same power or degree. The property of homogeneity simplifies the process of finding solutions, and plays a crucial role in identifying integrating factors for certain types of differential equations.

For example, the equation \((x^2 y - 2xy^2) \, dx - (x^3 - 3x^2 y) \, dy = 0\) is homogeneous because both terms \(x^2 y - 2xy^2\) and \(x^3 - 3x^2 y\) are of degree 3.

The degree corresponds to the sum of the exponents of the variables in each term. This characteristic often simplifies finding solutions, as certain techniques and transformations rely on the equation's homogeneity to work effectively. Homogeneous equations often allow for simplifying assumptions and substitutions, especially when combined with integrating factors that take advantage of the equation's structure.

Partial Derivatives in Differential Equations

Partial derivatives involve differentiating a function with respect to one variable while keeping others constant. In the context of differential equations, they're used to test for exactness. If a differential equation \( M dx + N dy = 0 \) is to be exact, then it must satisfy the condition: \( \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \).

• For instance, given \( M = x^2y - 2xy^2 \) and \( N = -(x^3 - 3x^2y) \), calculate \( \frac{\partial M}{\partial y} \) and \( \frac{\partial N}{\partial x} \).
• Upon computation: \( \frac{\partial M}{\partial y} = x^2 - 4xy \) and \( \frac{\partial N}{\partial x} = -3x^2 + 6xy \).


Since these derivatives are not equal, the equation isn't exact. However, partial derivatives guide us towards the right integrating factor or transformation to convert the equation to an exact form. This step is crucial as it directly influences the approach we take to solve the differential equation.