Clairaut’s Equation Solved: Key Concepts

Clairaut’s equation is a first-order differential equation of the form y = x rac{dy}{dx} + f(rac{dy}{dx}), where f is a differentiable function of the derivative alone. Unlike standard ODEs, this equation yields two types of solutions: a family of straight lines (general solution) and a singular curve (envelope). Understanding Clairaut’s equation solved is essential because:

• It appears in 5-6% of CSIR NET Mathematics questions, contributing 4-6 marks per problem.
• It simplifies complex problems by avoiding redundant integration steps.
• It bridges theory and application, useful in optics, mechanics, and chemical engineering.

VedPrep’s VedPrep offers tailored resources to help you master this topic efficiently. Watch this free VedPrep lecture to visualize the solution process step-by-step.

Step 1: Identify the Canonical Form of Clairaut’s equation solved

The general form is y = x rac{dy}{dx} + f(rac{dy}{dx}). Here, rac{dy}{dx} is treated as a parameter, say p, leading to the rewritten form:

y = x p + f(p)

For example, if f(p) = p^2, the equation becomes y = x p + p^2. This is a classic case of Clairaut’s equation solved where the general solution is a family of straight lines parameterized by p.

Step 2: Derive the General Solution

Treat p as a constant. The general solution is a family of straight lines:

y = C x + f(C), where C is an arbitrary constant representing p. Each line corresponds to a unique slope C.

For the example above, the general solution is y = C x + C^2. This family of parabolas (when f(p) = p^2) illustrates how Clairaut’s equation solved generates a one-parameter family of curves.

Step 3: Find the Singular Solution (Envelope)

The singular solution is the envelope of the family of curves. To find it:

1. Differentiate the general solution with respect to C:
2. Set the derivative equal to zero to find the condition for the envelope:
3. Solve for x in terms of C using x = -f'(C).
4. Substitute back into the general solution to eliminate C.

For f(p) = p^2, the envelope condition is x = -2C. Substituting back yields the singular solution:

y = -rac{x^2}{4}

This curve is the singular solution, which is not part of the general family but touches every member of it. It’s a critical concept in Clairaut’s equation solved that often appears in UPSC questions.

Step 4: Verify the Singular Solution

Substitute the singular solution back into the original Clairaut equation to confirm its validity. For y = -rac{x^2}{4}, verify that it satisfies y = x rac{dy}{dx} + (rac{dy}{dx})^2. This step is often overlooked but earns partial marks in exams.

In UPSC, always include this verification to demonstrate thorough understanding of Clairaut’s equation solved.

Step 5: Solve Practice Problems with Clairaut’s equation solved

Practice is key to mastering Clairaut’s equation solved. Here’s a sample problem:

Problem:

Solve the differential equation y = x rac{dy}{dx} + (rac{dy}{dx})^2. Identify the singular solution.

Solution:

1. Let p = rac{dy}{dx}. The equation becomes y = x p + p^2.
2. The general solution is y = C x + C^2.
3. Differentiate with respect to C to get x = -2C.
4. Substitute C = -rac{x}{2} into the general solution to obtain the singular solution:

y = -rac{x^2}{4}

The correct answer is A if the options include y = -rac{x^2}{4}.

Common Pitfalls in Clairaut’s equation solved

Many students confuse the singular solution with a particular solution. The singular solution is the envelope of the family of curves, not just another member. Here’s how to avoid mistakes:

• Misidentification: Treating the singular curve as a particular solution by fixing C in the general family.
• Incorrect Differentiation: Differentiating y = C x + f(C) with respect to x instead of C introduces errors.
• Sign Errors: Forgetting the negative sign in x = -f'(C) can lead to incorrect results.

Always double-check your steps when solving Clairaut’s equation solved problems.

Real-World Applications of Clairaut’s equation solved

Clairaut’s equation solved isn’t just theoretical—it has practical applications:

• Chemical Engineering: Models concentration profiles in plug-flow reactors, helping optimize catalyst loading.
• Aerospace: Predicts thin-film stress during deposition, ensuring durable coatings for turbine blades.
• Environmental Science: Calibrates gas-sensor arrays for reliable air-quality monitoring.

Understanding these applications deepens your grasp of Clairaut’s equation solved and its relevance to UPSC Optional exams.

How to Prepare for Clairaut’s equation solved in UPSC Optional

Follow this structured approach to ace Clairaut’s equation solved:

1. Memorize the Canonical Form: Know the structure y = x p + f(p) and recognize it instantly.
2. Practice Step-by-Step: Start with solved examples, then move to timed drills mixing Clairaut’s equation with other first-order ODEs.
3. Use VedPrep Resources: Access VedPrep’s video lessons, practice sheets, and solved papers tailored for UPSC Optional. Watch this lecture to see the algorithm in action.
4. Graphical Interpretation: Sketch the family of lines and their envelope to visualize the solution. This earns extra marks in UPSC.
5. Verify Every Solution: Always substitute the singular solution back into the original equation to confirm its validity.

For additional practice, explore VedPrep’s VedPrep platform, which offers comprehensive materials for UPSC Optional preparation.

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