Problem 43 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Solution of the equation \(x d y=\left(y+x \frac{f(y / x)}{f^{\prime}(y / x)}\right) d x\) is (A) \(f\left(\frac{x}{y}\right)=c y\) (B) \(f\left(\frac{y}{x}\right)=c x\) (C) \(f\left(\frac{y}{x}\right)=c x y\) (D) None of these

Step-by-Step Solution

Answer

(B) \(f\left(\frac{y}{x}\right)=c x\) is the solution.

Step 1: Rewrite the Given Equation

The given differential equation is \( x \frac{dy}{dx} = y + x \frac{f\left(\frac{y}{x}\right)}{f'\left(\frac{y}{x}\right)} \).Let's rewrite the equation to clarify the manipulation needed for further steps. Notice that we can separate differentials: \( x \, dy = \left( y + x \frac{f\left(\frac{y}{x}\right)}{f'\left(\frac{y}{x}\right)} \right) dx \).

Step 2: Substitute Variables

Introduce a substitution to simplify the equation: let \( v = \frac{y}{x} \) which implies \( y = vx \) and \( dy = v \, dx + x \, dv \).Substitute \( dy \) in the given equation: \[ x(v \, dx + x \, dv) = vx + x \frac{f(v)}{f'(v)} \, dx \].

Step 3: Simplify and Rearrange

Expand the left side: \[ xv \, dx + x^2 \, dv = vx \, dx + x \frac{f(v)}{f'(v)} \, dx \].Rearranging gives us: \[ x^2 \, dv = x \frac{f(v)}{f'(v)} \, dx \].This simplifies further to: \[ x \, dv = \frac{f(v)}{f'(v)} \, dx \].

Step 4: Integrate to Find General Solution

The differential equation \( x \, dv = \frac{f(v)}{f'(v)} \, dx \) can be separated: \[ dv = \frac{f(v)}{xf'(v)} \, dx \].Separate the variables to integrate both sides: \[ \int f'(v) \cdot dv = \int \frac{1}{x} \, dx \],which leads to \[ f(v) = \ln|x| + C \] where \( C \) is an integration constant.

Step 5: Interpret the Solution

Substitute back \( v = \frac{y}{x} \): \[ f\left( \frac{y}{x} \right) = \ln|x| + C \].This implies that any function of the form \( f\left( \frac{y}{x} \right) = c \cdot x \) satisfies the differential equation where \( c \) is a constant. Since this form matches option (B), the solution corresponds to that choice.

Key Concepts

Variable Substitution

When dealing with differential equations, variable substitution can be a powerful technique to simplify and resolve equations that might initially appear complex.
In our exercise, we employed substitution by introducing a new variable \( v = \frac{y}{x} \). This transforms the equation into terms of \( v \) and \( x \), making it potentially easier to separate and solve.

The benefit of this substitution is that it reduces the number of variables, consolidating them into one that can be managed more easily. Specifically, if \( y = vx \), then the differential \( dy \) becomes \( v \, dx + x \, dv \).

• This new representation allows us to swap out the dependence on \( y \) entirely in favor of \( v \) and its differentials.
• It simplifies the manipulation and often unveils an underlying simplicity in the structure of the problem.

In summary, variable substitution is particularly useful in simplifying differential equations when you notice expressions like \( \frac{y}{x} \) appear frequently, indicating that a substitution might lead to a general simplification.

Separation of Variables

Separation of variables is a method used to solve certain types of differential equations. It involves reorganizing the equation so that each variable is on a different side of the equation.
For the problem at hand, after substituting the variable \( v \), we reached an equation where \( dv \) and \( x \, dx \) could be separated as: \( dv = \frac{f(v)}{xf'(v)} \, dx \).
This allows both sides of the equation to be integrated separately:

• On one side, you integrate with respect to \( v \).
• On the other side, you integrate with respect to \( x \).

This leads to a general solution finding approach where each part is integrated separately. The core idea of separation lies in rearranging the equation such that the outcome can directly be integrated.
This isolation of variables often simplifies the integration process, making it easier to move towards a solution. It's a critical technique in calculus and differential equations for isolating variables effectively.

Integration Techniques

Integrating functions is often the final step in solving separated differential equations. Here, integration resolves statements into functions that can be interpreted as solutions.
In our exercise, once variables were successfully separated, each side of the equation was integrated.

We had:\[\int f'(v) \cdot dv = \int \frac{1}{x} \, dx \]
Performing these integrations leads to:

• \( f(v) \) when integrating with respect to \( v \).
• \( \ln|x| + C \) when integrating with respect to \( x \), where \( C \) is the integration constant.

The integration technique utilized here relies on common integration rules and an understanding that these solutions represent families of functions distinguished by arbitrary constants.
These constants emerge because the solution to a differential equation isn't a single function, but rather a collection of functions, each differing by the value of the constant. Integrating leads to a general expression that forms the cornerstone of the solution set.

Mastering integration techniques is essential since finding these antiderivatives underpins the ability to form meaningful functions from separated differential equations.