{"id":32656,"date":"2026-08-31T01:33:35","date_gmt":"2026-08-31T01:33:35","guid":{"rendered":"https:\/\/www.vedprep.com\/exams\/?p=32656"},"modified":"2026-08-31T01:33:35","modified_gmt":"2026-08-31T01:33:35","slug":"clairaut-s-equation-solved","status":"publish","type":"post","link":"https:\/\/www.vedprep.com\/exams\/upsc\/clairaut-s-equation-solved\/","title":{"rendered":"Clairaut\u2019s Equation Solved: 5 Proven Steps for UPSC"},"content":{"rendered":"<article>\n<header>\n<h1>Clairaut\u2019s Equation Solved: 5 Proven Steps for UPSC Optional Success<\/h1>\n<\/header>\n<div>\n<section>\n<p>UPSC Optional exams demand precision in mathematical problem-solving, and <strong>Clairaut\u2019s equation solved<\/strong> is a critical topic for aspirants preparing for subjects like Mathematics, Chemistry, and Physics. This guide breaks down the singular solution method into actionable steps, ensuring you master the concept for exams like CSIR NET, IIT JAM, and GATE.<\/p>\n<\/section>\n<h2>Clairaut\u2019s Equation Solved: Key Concepts<\/h2>\n<p>Clairaut\u2019s equation is a first-order differential equation of the form <code>y = x rac{dy}{dx} + f(rac{dy}{dx})<\/code>, where <code>f<\/code> 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 <strong>Clairaut\u2019s equation solved<\/strong> is essential because:<\/p>\n<ul>\n<li>It appears in <strong>5-6% of CSIR NET Mathematics questions<\/strong>, contributing 4-6 marks per problem.<\/li>\n<li>It simplifies complex problems by avoiding redundant integration steps.<\/li>\n<li>It bridges theory and application, useful in optics, mechanics, and chemical engineering.<\/li>\n<\/ul>\n<p>VedPrep\u2019s <a href=\"https:\/\/www.vedprep.com\/\">VedPrep<\/a> offers tailored resources to help you master this topic efficiently. Watch this <a href=\"https:\/\/www.youtube.com\/watch?v=kL4iszkKnqg\" target=\"_blank\" rel=\"noopener nofollow\">free VedPrep lecture<\/a> to visualize the solution process step-by-step.<\/p>\n<h2>Step 1: Identify the Canonical Form of <strong>Clairaut\u2019s equation solved<\/strong><\/h2>\n<p>The general form is <code>y = x rac{dy}{dx} + f(rac{dy}{dx})<\/code>. Here, <code>rac{dy}{dx}<\/code> is treated as a parameter, say <code>p<\/code>, leading to the rewritten form:<\/p>\n<p><code>y = x p + f(p)<\/code><\/p>\n<p>For example, if <code>f(p) = p^2<\/code>, the equation becomes <code>y = x p + p^2<\/code>. This is a classic case of <strong>Clairaut\u2019s equation solved<\/strong> where the general solution is a family of straight lines parameterized by <code>p<\/code>.<\/p>\n<h2>Step 2: Derive the General Solution<\/h2>\n<p>Treat <code>p<\/code> as a constant. The general solution is a family of straight lines:<\/p>\n<p><code>y = C x + f(C)<\/code>, where <code>C<\/code> is an arbitrary constant representing <code>p<\/code>. Each line corresponds to a unique slope <code>C<\/code>.<\/p>\n<p>For the example above, the general solution is <code>y = C x + C^2<\/code>. This family of parabolas (when <code>f(p) = p^2<\/code>) illustrates how <strong>Clairaut\u2019s equation solved<\/strong> generates a one-parameter family of curves.<\/p>\n<h2>Step 3: Find the Singular Solution (Envelope)<\/h2>\n<p>The singular solution is the envelope of the family of curves. To find it:<\/p>\n<ol>\n<li>Differentiate the general solution with respect to <code>C<\/code>:<\/li>\n<li>Set the derivative equal to zero to find the condition for the envelope:<\/li>\n<li>Solve for <code>x<\/code> in terms of <code>C<\/code> using <code>x = -f'(C)<\/code>.<\/li>\n<li>Substitute back into the general solution to eliminate <code>C<\/code>.<\/li>\n<\/ol>\n<p>For <code>f(p) = p^2<\/code>, the envelope condition is <code>x = -2C<\/code>. Substituting back yields the singular solution:<\/p>\n<p><code>y = -rac{x^2}{4}<\/code><\/p>\n<p>This curve is the singular solution, which is not part of the general family but touches every member of it. It\u2019s a critical concept in <strong>Clairaut\u2019s equation solved<\/strong> that often appears in UPSC questions.<\/p>\n<h2>Step 4: Verify the Singular Solution<\/h2>\n<p>Substitute the singular solution back into the original Clairaut equation to confirm its validity. For <code>y = -rac{x^2}{4}<\/code>, verify that it satisfies <code>y = x rac{dy}{dx} + (rac{dy}{dx})^2<\/code>. This step is often overlooked but earns partial marks in exams.<\/p>\n<p>In UPSC, always include this verification to demonstrate thorough understanding of <strong>Clairaut\u2019s equation solved<\/strong>.<\/p>\n<h2>Step 5: Solve Practice Problems with <strong>Clairaut\u2019s equation solved<\/strong><\/h2>\n<p>Practice is key to mastering <strong>Clairaut\u2019s equation solved<\/strong>. Here\u2019s a sample problem:<\/p>\n<h3>Problem:<\/h3>\n<p>Solve the differential equation <code>y = x rac{dy}{dx} + (rac{dy}{dx})^2<\/code>. Identify the singular solution.<\/p>\n<h3>Solution:<\/h3>\n<ol>\n<li>Let <code>p = rac{dy}{dx}<\/code>. The equation becomes <code>y = x p + p^2<\/code>.<\/li>\n<li>The general solution is <code>y = C x + C^2<\/code>.<\/li>\n<li>Differentiate with respect to <code>C<\/code> to get <code>x = -2C<\/code>.<\/li>\n<li>Substitute <code>C = -rac{x}{2}<\/code> into the general solution to obtain the singular solution:<\/li>\n<p><code>y = -rac{x^2}{4}<\/code><\/p>\n<\/ol>\n<p>The correct answer is <strong>A<\/strong> if the options include <code>y = -rac{x^2}{4}<\/code>.<\/p>\n<h2>Common Pitfalls in <strong>Clairaut\u2019s equation solved<\/strong><\/h2>\n<p>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\u2019s how to avoid mistakes:<\/p>\n<ul>\n<li><strong>Misidentification:<\/strong> Treating the singular curve as a particular solution by fixing <code>C<\/code> in the general family.<\/li>\n<li><strong>Incorrect Differentiation:<\/strong> Differentiating <code>y = C x + f(C)<\/code> with respect to <code>x<\/code> instead of <code>C<\/code> introduces errors.<\/li>\n<li><strong>Sign Errors:<\/strong> Forgetting the negative sign in <code>x = -f'(C)<\/code> can lead to incorrect results.<\/li>\n<\/ul>\n<p>Always double-check your steps when solving <strong>Clairaut\u2019s equation solved<\/strong> problems.<\/p>\n<h2>Real-World Applications of <strong>Clairaut\u2019s equation solved<\/strong><\/h2>\n<p><strong>Clairaut\u2019s equation solved<\/strong> isn\u2019t just theoretical\u2014it has practical applications:<\/p>\n<ul>\n<li><strong>Chemical Engineering:<\/strong> Models concentration profiles in plug-flow reactors, helping optimize catalyst loading.<\/li>\n<li><strong>Aerospace:<\/strong> Predicts thin-film stress during deposition, ensuring durable coatings for turbine blades.<\/li>\n<li><strong>Environmental Science:<\/strong> Calibrates gas-sensor arrays for reliable air-quality monitoring.<\/li>\n<\/ul>\n<p>Understanding these applications deepens your grasp of <strong>Clairaut\u2019s equation solved<\/strong> and its relevance to UPSC Optional exams.<\/p>\n<h2>How to Prepare for <strong>Clairaut\u2019s equation solved<\/strong> in UPSC Optional<\/h2>\n<p>Follow this structured approach to ace <strong>Clairaut\u2019s equation solved<\/strong>:<\/p>\n<ol>\n<li><strong>Memorize the Canonical Form:<\/strong> Know the structure <code>y = x p + f(p)<\/code> and recognize it instantly.<\/li>\n<li><strong>Practice Step-by-Step:<\/strong> Start with solved examples, then move to timed drills mixing Clairaut\u2019s equation with other first-order ODEs.<\/li>\n<li><strong>Use VedPrep Resources:<\/strong> Access VedPrep\u2019s video lessons, practice sheets, and solved papers tailored for UPSC Optional. <a href=\"https:\/\/www.youtube.com\/watch?v=kL4iszkKnqg\" target=\"_blank\" rel=\"noopener nofollow\">Watch this lecture<\/a> to see the algorithm in action.<\/li>\n<li><strong>Graphical Interpretation:<\/strong> Sketch the family of lines and their envelope to visualize the solution. This earns extra marks in UPSC.<\/li>\n<li><strong>Verify Every Solution:<\/strong> Always substitute the singular solution back into the original equation to confirm its validity.<\/li>\n<\/ol>\n<p>For additional practice, explore VedPrep\u2019s <a href=\"https:\/\/www.vedprep.com\/\">VedPrep<\/a> platform, which offers comprehensive materials for UPSC Optional preparation.<\/p>\n<h2>FAQs on <strong>Clairaut\u2019s equation solved<\/strong><\/h2>\n<section class=\"vedprep-faq\">\n<h3>Core Understanding<\/h3>\n<div class=\"faq-item\">\n<h4>What is the canonical form of <strong>Clairaut\u2019s equation solved<\/strong>?<\/h4>\n<p>The canonical form is <code>y = x rac{dy}{dx} + f(rac{dy}{dx})<\/code>, where <code>f<\/code> is a differentiable function of the derivative alone.<\/p>\n<\/p><\/div>\n<div class=\"faq-item\">\n<h4>How do you derive the general solution?<\/h4>\n<p>Treat <code>rac{dy}{dx}<\/code> as a constant parameter <code>p<\/code>. Rewrite the equation as <code>y = x p + f(p)<\/code>. The general solution is the family of straight lines <code>y = C x + f(C)<\/code>, where <code>C<\/code> is an arbitrary constant.<\/p>\n<\/p><\/div>\n<div class=\"faq-item\">\n<h4>What is the singular solution in <strong>Clairaut\u2019s equation solved<\/strong>?<\/h4>\n<p>The singular solution is the envelope of the family of curves, obtained by eliminating the parameter <code>C<\/code> between <code>y = C x + f(C)<\/code> and <code>rac{dy}{dC} = x + f'(C) = 0<\/code>. It\u2019s not part of the general family but touches every member.<\/p>\n<\/p><\/div>\n<div class=\"faq-item\">\n<h4>Why is <strong>Clairaut\u2019s equation solved<\/strong> classified as a first-order ODE?<\/h4>\n<p>It involves only the first derivative <code>rac{dy}{dx}<\/code> and no higher-order derivatives. The equation is solved by treating the derivative as a constant parameter.<\/p>\n<\/p><\/div>\n<div class=\"faq-item\">\n<h4>Can <strong>Clairaut\u2019s equation solved<\/strong> have multiple singular solutions?<\/h4>\n<p>Typically, it yields a single singular solution. However, if <code>f(p)<\/code> is not strictly convex, the envelope may consist of multiple branches.<\/p>\n<\/p><\/div>\n<\/section>\n<section class=\"vedprep-faq\">\n<h3>Exam Application<\/h3>\n<div class=\"faq-item\">\n<h4>How is <strong>Clairaut\u2019s equation solved<\/strong> tested in UPSC Optional?<\/h4>\n<p>UPSC tests your ability to derive general and singular solutions, identify envelope curves, and apply the concept to physical problems. Expect 4-6 marks per problem, with clear steps and concise final expressions.<\/p>\n<\/p><\/div>\n<div class=\"faq-item\">\n<h4>What shortcut helps solve <strong>Clairaut\u2019s equation solved<\/strong> quickly?<\/h4>\n<p>Treat <code>rac{dy}{dx}<\/code> as a constant parameter <code>p<\/code>, write <code>y = x p + f(p)<\/code>, and directly write the family <code>y = C x + f(C)<\/code>. Differentiate with respect to <code>C<\/code> to find the envelope condition <code>x = -f'(C)<\/code>.<\/p>\n<\/p><\/div>\n<\/section>\n<section class=\"vedprep-faq\">\n<h3>Common Mistakes<\/h3>\n<div class=\"faq-item\">\n<h4>Why do students confuse singular and particular solutions?<\/h4>\n<p>Students often mistake the singular solution for a particular solution because both are specific curves. The singular solution is an envelope, not obtained by fixing the constant in the general family.<\/p>\n<\/p><\/div>\n<div class=\"faq-item\">\n<h4>How to avoid sign errors in the envelope condition?<\/h4>\n<p>Remember the condition <code>x = -f'(C)<\/code> arises from <code>rac{dy}{dC} = 0<\/code>. Keep the negative sign explicit to prevent errors during substitution.<\/p>\n<\/p><\/div>\n<\/section>\n<section class=\"vedprep-faq\">\n<h3>Advanced Concepts<\/h3>\n<div class=\"faq-item\">\n<h4>How does <strong>Clairaut\u2019s equation solved<\/strong> relate to the method of characteristics?<\/h4>\n<p>Clairaut\u2019s equation can be viewed as a first-order PDE reduced to ODE form. The characteristic curves are the straight-line solutions, and the singular solution represents the envelope where characteristics intersect.<\/p>\n<\/p><\/div>\n<div class=\"faq-item\">\n<h4>What is the geometric interpretation of the singular solution?<\/h4>\n<p>The singular solution is the envelope curve tangent to every member of the family of straight lines. It represents the locus of points where the family\u2019s slope changes continuously.<\/p>\n<\/p><\/div>\n<\/section>\n<\/div>\n<\/article>\n","protected":false},"excerpt":{"rendered":"<p>In this guide we dissect Clairaut\u2019s equation, explaining its general and singular solutions with step\u2011by\u2011step derivations. We then explore common exam questions from CSIR NET, IIT JAM, and GATE, providing worked examples and practice problems to solidify your understanding today.<\/p>\n","protected":false},"author":12,"featured_media":32655,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":"","_debug_hook_fired":"2026-08-31 01:33:36","rank_math_seo_score":0},"categories":[353],"tags":[25817,25818,25820,25819,2923,2922],"class_list":["post-32656","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-upsc","tag-clairaut-s-equation-and-singular-solution-for-upsc-civil-services-optional-subjects","tag-clairaut-s-equation-and-singular-solution-for-upsc-civil-services-optional-subjects-notes","tag-clairaut-s-equation-and-singular-solution-for-upsc-civil-services-optional-subjects-practice","tag-clairaut-s-equation-and-singular-solution-for-upsc-civil-services-optional-subjects-questions","tag-competitive-exams","tag-vedprep","entry","has-media"],"acf":[],"rank_math_title":"Clairaut\u2019s Equation Solved: 5 Proven Steps for UPSC","rank_math_description":"Clairaut\u2019s equation solved: Master the singular solution for UPSC Optional exams with VedPrep\u2019s expert guide. Ace CSIR NET, IIT JAM, and GATE with confidence.","rank_math_focus_keyword":"Clairaut\u2019s equation solved","_links":{"self":[{"href":"https:\/\/www.vedprep.com\/exams\/wp-json\/wp\/v2\/posts\/32656","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.vedprep.com\/exams\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.vedprep.com\/exams\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.vedprep.com\/exams\/wp-json\/wp\/v2\/users\/12"}],"replies":[{"embeddable":true,"href":"https:\/\/www.vedprep.com\/exams\/wp-json\/wp\/v2\/comments?post=32656"}],"version-history":[{"count":1,"href":"https:\/\/www.vedprep.com\/exams\/wp-json\/wp\/v2\/posts\/32656\/revisions"}],"predecessor-version":[{"id":35538,"href":"https:\/\/www.vedprep.com\/exams\/wp-json\/wp\/v2\/posts\/32656\/revisions\/35538"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.vedprep.com\/exams\/wp-json\/wp\/v2\/media\/32655"}],"wp:attachment":[{"href":"https:\/\/www.vedprep.com\/exams\/wp-json\/wp\/v2\/media?parent=32656"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.vedprep.com\/exams\/wp-json\/wp\/v2\/categories?post=32656"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.vedprep.com\/exams\/wp-json\/wp\/v2\/tags?post=32656"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}