Problem 48 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The solution of the differential equation \(x^{2} \frac{d y}{d x}-x y=1+\cos \frac{y}{x}\) is (A) \(\cos \frac{y}{x}=1+\frac{c}{x}\) (B) \(x^{2}=\left(c+x^{2}\right) \tan \frac{y}{x}\) (C) \(\tan \frac{y}{2 x}=c-\frac{1}{2 x^{2}}\) (D) \(\tan \frac{y}{x}=c+\frac{1}{x}\)

Step-by-Step Solution

Answer

The correct option is (C): \(\tan \frac{y}{2 x}=c-\frac{1}{2 x^{2}}\).

Step 1: Identify the Type of Differential Equation

The given differential equation is \( x^{2} \frac{dy}{dx} - xy = 1 + \cos \frac{y}{x} \). This equation appears to have variables separable in a non-standard form.

Step 2: Substitution for Simplicity

Introduce a substitution to simplify the equation. Let \( v = \frac{y}{x} \), then \( y = vx \) and \( \frac{dy}{dx} = v + x \frac{dv}{dx} \). Substitute into the original equation.

Step 3: Substitute and Simplify

Substitute \( y = vx \) and \( \frac{dy}{dx} = v + x \frac{dv}{dx} \) into the equation:\[ x^{2}(v + x \frac{dv}{dx}) - x(vx) = 1 + \cos v \]This simplifies to:\[ x^{3} \frac{dv}{dx} = 1 + \cos v \]

Step 4: Separate Variables

Now, separate variables by rewriting the equation as:\[ \frac{dv}{1 + \cos v} = \frac{dx}{x^{3}} \]This separates the variables, allowing for integration on both sides.

Step 5: Integrate Both Sides

Integrate both sides:The left side:\[ \int \frac{dv}{1 + \cos v} \]The right side:\[ \int \frac{dx}{x^{3}} \]The integration results are obtained through specific formulas or substitutions: Left hand side uses the identity \( 1+\cos v = 2\cos^2\frac{v}{2} \) and substitution, right hand side is \( \frac{-1}{2x^{2}} + C \).

Step 6: Solve the Integrals

Using the trigonometric identity:For the left side: \( \int \frac{dv}{1 + \cos v} = \int \frac{dv}{2 \cos^2\frac{v}{2}} = \tan\frac{v}{2} + C_1 \)For the right side: \( \int \frac{dx}{x^3} = \frac{-1}{2x^2} + C_2 \)

Step 7: Combine the Integration Results

Equate the integration results to obtain the general solution:\[ \tan\frac{v}{2} = \frac{-1}{2x^2} + C \]where \( C = C_2 - C_1 \). Substitute back \( v = \frac{y}{x} \).

Step 8: Choose the Correct Answer

Substitute \( v = \frac{y}{x} \) back into the result:\[ \tan\frac{y}{2x} = C - \frac{1}{2x^2} \]This matches option (C) given in the problem.

Key Concepts

Separation of Variables

Separation of variables is a strategy used to solve differential equations where variables can be placed on opposite sides of the equation. The aim is to regroup the terms of a differential equation so that each side contains only one variable and its differential. This generally involves rewriting the equation in a form where one can isolate and integrate each variable separately.

• First, identify if the equation can be rearranged so that all terms containing one variable and its differential can be on one side and the rest on the other.
• After separation, each side of the equation becomes a function of one variable, allowing us to integrate each side independently.
• This method simplifies the solving process significantly when applicable, because it reduces the problem to simple integration tasks.

In our current problem, we separated \( \frac{dv}{1 + \cos v} \) on one side and \( \frac{dx}{x^{3}} \) on the other. Through integration, we derive each variable's interaction within the equation.

Trigonometric Identities

Trigonometric identities are essential tools in calculus, especially when dealing with equations involving trigonometric functions. They help simplify complex expressions and are often used to facilitate integration or differentiation.

• One common identity used in our problem is the cosine double-angle identity: \( 1 + \cos v = 2 \cos^2 \frac{v}{2} \). This identity transforms the expression into a more manageable form for integration purposes.
• Understanding and applying these identities allows integrals or derivatives of trigonometric functions to be calculated more easily, which is crucial in finding solutions to differential equations involving trigonometric functions.

By reformulating the integral \( \int \frac{dv}{1 + \cos v} \) in terms of \( \cos^2 \frac{v}{2} \), the integration becomes straightforward, leading to the solution form involving a tangent function.

Substitution Method

The substitution method simplifies differential equations by expressing one variable in terms of another. Substitution changes the variables in an equation, making it easier to work with.

• First, identify a substitution that simplifies the equation's structure, often by expressing one variable as a function of another, such as \( v = \frac{y}{x} \).
• This substitution not only changes the original equation but also changes the differentiation, which must be accounted for. In our example, \( y = vx \) and \( \frac{dy}{dx} = v + x \frac{dv}{dx} \) after substitution.
• The transformed equation should allow for clearer paths to separate variables or resolve the stored equation, leading to a more straightforward solution.

In this exercise, substitution made the complex differential equation more manageable and amenable to the separation of variables approach. After substituting, we could isolate each variable's contributions, leading to a successful integration to solve the equation.