JEE Exam > JEE Notes > Mathematics (Maths) Main & Advanced > Differential Equation Solved Examples
Differential Equation Solved Examples
| Download, print and study this document offline |
Please wait while the PDF view is loading
Page 1 Solved Examples on Differential Equation JEE Main Single Correct Type Q1. The order and degree of the differential equation ?? = ?? ???? ???? + v ?? ?? ( ???? ???? ) ?? + ?? ?? are (a) 1,2 (b) 2,1 (c) 1,1 (d) 2,2 Ans: (a) Clearly, highest order derivative involved is ???? ???? , having order 1 . Expressing the above differential equation as a polynomial in derivative, we have ( ?? - ?? ???? ???? ) 2 = ?? 2 ( ???? ???? ) 2 + ?? 2 i.e., ( ?? 2 - ?? 2 ) ( ???? ???? ) 2 - 2 ???? ???? ???? + ?? 2 - ?? 2 = 0 In this equation, the power of highest order derivative is 2 . So its degree is 2 . Q2. The degree of the differential equation ?? ?? ?? ?? ?? ?? + ?? ( ???? ???? ) ?? = ?? ?? ?????? ? ( ?? ?? ?? ?? ?? ?? ) is (a) 1 (b) 2 (c) 3 (d) None of these Ans: (d) The above equation cannot be written as a polynomial in derivatives due to the term ?? 2 log ? ( ?? 2 ?? ?? ?? 2 ). Hence degree of the differential equation is 'not defined'. Page 2 Solved Examples on Differential Equation JEE Main Single Correct Type Q1. The order and degree of the differential equation ?? = ?? ???? ???? + v ?? ?? ( ???? ???? ) ?? + ?? ?? are (a) 1,2 (b) 2,1 (c) 1,1 (d) 2,2 Ans: (a) Clearly, highest order derivative involved is ???? ???? , having order 1 . Expressing the above differential equation as a polynomial in derivative, we have ( ?? - ?? ???? ???? ) 2 = ?? 2 ( ???? ???? ) 2 + ?? 2 i.e., ( ?? 2 - ?? 2 ) ( ???? ???? ) 2 - 2 ???? ???? ???? + ?? 2 - ?? 2 = 0 In this equation, the power of highest order derivative is 2 . So its degree is 2 . Q2. The degree of the differential equation ?? ?? ?? ?? ?? ?? + ?? ( ???? ???? ) ?? = ?? ?? ?????? ? ( ?? ?? ?? ?? ?? ?? ) is (a) 1 (b) 2 (c) 3 (d) None of these Ans: (d) The above equation cannot be written as a polynomial in derivatives due to the term ?? 2 log ? ( ?? 2 ?? ?? ?? 2 ). Hence degree of the differential equation is 'not defined'. Q3. Differential equation whose general solution is ?? = ?? ?? ?? + ?? ?? ?? for all values of ?? ?? and ?? ?? is (a) ?? ?? ?? ?? ?? ?? + ?? ?? ?? + ???? ???? = ?? (b) ?? ?? ?? ?? ?? ?? + ?? ?? ?? - ???? ???? = ?? (c) ?? ?? ?? ?? ?? ?? - ?? ?? ?? ???? ???? = ?? (d) ?? ?? ?? ?? ?? ?? + ?? ?? ???? ???? - ?? ?? ?? = ?? Ans: (d) ?? = ?? 1 ?? + ?? 2 ?? (i) There are two arbitrary constants. To eliminate these constants, we need to differentiate (i) twice. Differentiating (i) with respect to ?? , ???? ???? = ?? 1 - ?? 2 ?? 2 Again differentiating with respect to ?? , ?? 2 ?? ?? ?? 2 = 2 ?? 2 ?? 3 From (iii), ?? 2 = ?? 3 2 ?? 2 ?? ?? ?? 2 and from (ii), ?? 1 = ???? ???? + ?? 2 ?? 2 ; ? ?? 1 = ???? ???? + ?? 2 ?? 2 ?? ?? ?? 2 From (i), ?? = ( ???? ???? + ?? 2 · ?? 2 ?? ?? ?? 2 ) ?? + ?? 2 2 · ?? 2 ?? ?? ?? 2 ? ?? = ?? 2 ?? 2 ?? ?? ?? 2 + ?? ???? ???? ? ?? 2 ?? ?? ?? 2 + 1 ?? ???? ???? - ?? ?? 2 = 0 Q4. ?? = ?? ?? + ?? is a solution of the differential equation (a) ?? ?? ???? ???? = ?? ?? (b) ?? ?? ???? ???? = ?? ?? (c) ?? ???? ???? = ?? (d) ?? ???? ???? = ?? Ans: (b) We have ?? = ?? ?? + 1 ? 1 ?? = ?? + 1 ?? = 1 + 1 ?? Differentiating w.r.t. ? - 1 ?? 2 ???? ???? = 0 - 1 ?? 2 ? ? ?? 2 ???? ???? = ?? 2 Page 3 Solved Examples on Differential Equation JEE Main Single Correct Type Q1. The order and degree of the differential equation ?? = ?? ???? ???? + v ?? ?? ( ???? ???? ) ?? + ?? ?? are (a) 1,2 (b) 2,1 (c) 1,1 (d) 2,2 Ans: (a) Clearly, highest order derivative involved is ???? ???? , having order 1 . Expressing the above differential equation as a polynomial in derivative, we have ( ?? - ?? ???? ???? ) 2 = ?? 2 ( ???? ???? ) 2 + ?? 2 i.e., ( ?? 2 - ?? 2 ) ( ???? ???? ) 2 - 2 ???? ???? ???? + ?? 2 - ?? 2 = 0 In this equation, the power of highest order derivative is 2 . So its degree is 2 . Q2. The degree of the differential equation ?? ?? ?? ?? ?? ?? + ?? ( ???? ???? ) ?? = ?? ?? ?????? ? ( ?? ?? ?? ?? ?? ?? ) is (a) 1 (b) 2 (c) 3 (d) None of these Ans: (d) The above equation cannot be written as a polynomial in derivatives due to the term ?? 2 log ? ( ?? 2 ?? ?? ?? 2 ). Hence degree of the differential equation is 'not defined'. Q3. Differential equation whose general solution is ?? = ?? ?? ?? + ?? ?? ?? for all values of ?? ?? and ?? ?? is (a) ?? ?? ?? ?? ?? ?? + ?? ?? ?? + ???? ???? = ?? (b) ?? ?? ?? ?? ?? ?? + ?? ?? ?? - ???? ???? = ?? (c) ?? ?? ?? ?? ?? ?? - ?? ?? ?? ???? ???? = ?? (d) ?? ?? ?? ?? ?? ?? + ?? ?? ???? ???? - ?? ?? ?? = ?? Ans: (d) ?? = ?? 1 ?? + ?? 2 ?? (i) There are two arbitrary constants. To eliminate these constants, we need to differentiate (i) twice. Differentiating (i) with respect to ?? , ???? ???? = ?? 1 - ?? 2 ?? 2 Again differentiating with respect to ?? , ?? 2 ?? ?? ?? 2 = 2 ?? 2 ?? 3 From (iii), ?? 2 = ?? 3 2 ?? 2 ?? ?? ?? 2 and from (ii), ?? 1 = ???? ???? + ?? 2 ?? 2 ; ? ?? 1 = ???? ???? + ?? 2 ?? 2 ?? ?? ?? 2 From (i), ?? = ( ???? ???? + ?? 2 · ?? 2 ?? ?? ?? 2 ) ?? + ?? 2 2 · ?? 2 ?? ?? ?? 2 ? ?? = ?? 2 ?? 2 ?? ?? ?? 2 + ?? ???? ???? ? ?? 2 ?? ?? ?? 2 + 1 ?? ???? ???? - ?? ?? 2 = 0 Q4. ?? = ?? ?? + ?? is a solution of the differential equation (a) ?? ?? ???? ???? = ?? ?? (b) ?? ?? ???? ???? = ?? ?? (c) ?? ???? ???? = ?? (d) ?? ???? ???? = ?? Ans: (b) We have ?? = ?? ?? + 1 ? 1 ?? = ?? + 1 ?? = 1 + 1 ?? Differentiating w.r.t. ? - 1 ?? 2 ???? ???? = 0 - 1 ?? 2 ? ? ?? 2 ???? ???? = ?? 2 Q5. The differential equation of family of curves whose tangent form an angle of ?? / ?? with the hyperbola ???? = ?? ?? is (a) ???? ???? = ?? ?? + ?? ?? ?? ?? - ?? ?? (b) ???? ???? = ?? ?? - ?? ?? ?? ?? + ?? ?? (c) ???? ???? = - ?? ?? ?? ?? (d) None of these Ans: (b) The slope of the tangent to the family of curves is ?? 1 = ???? ???? Equation of the hyperbola is ???? = ?? 2 ? ?? = ?? 2 ?? ? ???? ???? = - ?? 2 ?? 2 ? Slope of tangent to ???? = ?? 2 is ?? 2 = - ?? 2 ?? 2 Now tan ? ?? 4 = ?? 1 - ?? 2 1 + ?? 1 ?? 2 ? 1 = ???? ???? + ?? 2 ?? 2 1 - ?? 2 ?? 2 ???? ???? ? ???? ???? ( 1 + ?? 2 ?? 2 ) = ( 1 - ?? 2 ?? 2 ) ? ???? ???? = ?? 2 - ?? 2 ?? 2 + ?? 2 Q5. The solution of the differential equation ( ?? + ?? ?? ) ???? ???? = ?? ( ?? + ?? ?? ) is (a) ?? ???? ?? - ?? ? ?? = ?????? ? ( ?? + ?? ?? ) + ?? (b) ???? ?? - ?? ? ?? = ?????? ? ( ?? + ?? ?? ) + ?? (c) ?? ???? ?? - ?? ? ?? + ?????? ? ( ?? + ?? ?? ) + ?? = ?? (d) None of these Ans: (a) Separating the variables, we can re-write the given differential equation as ???? ?? 1 + ?? 2 = ???? 1 + ?? 2 ? ? ? 2 ???? ?? 1 + ?? 2 = 2 ? ? ???? 1 + ?? 2 ? 2 tan - 1 ? ?? = log ?? ? ( 1 + ?? 2 ) + ?? Q6. The solution of ???? ???? = ?? ?? ?? + ?? ???? ? ?? is (a) ?? = ?? ?? ?? - ?? ???? ? ?? + ?? (b) ?? + ?? ???? ? ?? = ?? + ?? (c) ?? = ?? ?? ?? + ?? ???? ? ?? + ?? (d) None of these Ans: (a) Given equation may be re -written as ???? = ( ?? 2 + sin ? ?? ) ???? Integrating, ? ? ???? = ? ? ( ?? 2 + sin ? ?? ) ???? ? ? ?? = ?? 3 3 - c os ? ?? + ?? Page 4 Solved Examples on Differential Equation JEE Main Single Correct Type Q1. The order and degree of the differential equation ?? = ?? ???? ???? + v ?? ?? ( ???? ???? ) ?? + ?? ?? are (a) 1,2 (b) 2,1 (c) 1,1 (d) 2,2 Ans: (a) Clearly, highest order derivative involved is ???? ???? , having order 1 . Expressing the above differential equation as a polynomial in derivative, we have ( ?? - ?? ???? ???? ) 2 = ?? 2 ( ???? ???? ) 2 + ?? 2 i.e., ( ?? 2 - ?? 2 ) ( ???? ???? ) 2 - 2 ???? ???? ???? + ?? 2 - ?? 2 = 0 In this equation, the power of highest order derivative is 2 . So its degree is 2 . Q2. The degree of the differential equation ?? ?? ?? ?? ?? ?? + ?? ( ???? ???? ) ?? = ?? ?? ?????? ? ( ?? ?? ?? ?? ?? ?? ) is (a) 1 (b) 2 (c) 3 (d) None of these Ans: (d) The above equation cannot be written as a polynomial in derivatives due to the term ?? 2 log ? ( ?? 2 ?? ?? ?? 2 ). Hence degree of the differential equation is 'not defined'. Q3. Differential equation whose general solution is ?? = ?? ?? ?? + ?? ?? ?? for all values of ?? ?? and ?? ?? is (a) ?? ?? ?? ?? ?? ?? + ?? ?? ?? + ???? ???? = ?? (b) ?? ?? ?? ?? ?? ?? + ?? ?? ?? - ???? ???? = ?? (c) ?? ?? ?? ?? ?? ?? - ?? ?? ?? ???? ???? = ?? (d) ?? ?? ?? ?? ?? ?? + ?? ?? ???? ???? - ?? ?? ?? = ?? Ans: (d) ?? = ?? 1 ?? + ?? 2 ?? (i) There are two arbitrary constants. To eliminate these constants, we need to differentiate (i) twice. Differentiating (i) with respect to ?? , ???? ???? = ?? 1 - ?? 2 ?? 2 Again differentiating with respect to ?? , ?? 2 ?? ?? ?? 2 = 2 ?? 2 ?? 3 From (iii), ?? 2 = ?? 3 2 ?? 2 ?? ?? ?? 2 and from (ii), ?? 1 = ???? ???? + ?? 2 ?? 2 ; ? ?? 1 = ???? ???? + ?? 2 ?? 2 ?? ?? ?? 2 From (i), ?? = ( ???? ???? + ?? 2 · ?? 2 ?? ?? ?? 2 ) ?? + ?? 2 2 · ?? 2 ?? ?? ?? 2 ? ?? = ?? 2 ?? 2 ?? ?? ?? 2 + ?? ???? ???? ? ?? 2 ?? ?? ?? 2 + 1 ?? ???? ???? - ?? ?? 2 = 0 Q4. ?? = ?? ?? + ?? is a solution of the differential equation (a) ?? ?? ???? ???? = ?? ?? (b) ?? ?? ???? ???? = ?? ?? (c) ?? ???? ???? = ?? (d) ?? ???? ???? = ?? Ans: (b) We have ?? = ?? ?? + 1 ? 1 ?? = ?? + 1 ?? = 1 + 1 ?? Differentiating w.r.t. ? - 1 ?? 2 ???? ???? = 0 - 1 ?? 2 ? ? ?? 2 ???? ???? = ?? 2 Q5. The differential equation of family of curves whose tangent form an angle of ?? / ?? with the hyperbola ???? = ?? ?? is (a) ???? ???? = ?? ?? + ?? ?? ?? ?? - ?? ?? (b) ???? ???? = ?? ?? - ?? ?? ?? ?? + ?? ?? (c) ???? ???? = - ?? ?? ?? ?? (d) None of these Ans: (b) The slope of the tangent to the family of curves is ?? 1 = ???? ???? Equation of the hyperbola is ???? = ?? 2 ? ?? = ?? 2 ?? ? ???? ???? = - ?? 2 ?? 2 ? Slope of tangent to ???? = ?? 2 is ?? 2 = - ?? 2 ?? 2 Now tan ? ?? 4 = ?? 1 - ?? 2 1 + ?? 1 ?? 2 ? 1 = ???? ???? + ?? 2 ?? 2 1 - ?? 2 ?? 2 ???? ???? ? ???? ???? ( 1 + ?? 2 ?? 2 ) = ( 1 - ?? 2 ?? 2 ) ? ???? ???? = ?? 2 - ?? 2 ?? 2 + ?? 2 Q5. The solution of the differential equation ( ?? + ?? ?? ) ???? ???? = ?? ( ?? + ?? ?? ) is (a) ?? ???? ?? - ?? ? ?? = ?????? ? ( ?? + ?? ?? ) + ?? (b) ???? ?? - ?? ? ?? = ?????? ? ( ?? + ?? ?? ) + ?? (c) ?? ???? ?? - ?? ? ?? + ?????? ? ( ?? + ?? ?? ) + ?? = ?? (d) None of these Ans: (a) Separating the variables, we can re-write the given differential equation as ???? ?? 1 + ?? 2 = ???? 1 + ?? 2 ? ? ? 2 ???? ?? 1 + ?? 2 = 2 ? ? ???? 1 + ?? 2 ? 2 tan - 1 ? ?? = log ?? ? ( 1 + ?? 2 ) + ?? Q6. The solution of ???? ???? = ?? ?? ?? + ?? ???? ? ?? is (a) ?? = ?? ?? ?? - ?? ???? ? ?? + ?? (b) ?? + ?? ???? ? ?? = ?? + ?? (c) ?? = ?? ?? ?? + ?? ???? ? ?? + ?? (d) None of these Ans: (a) Given equation may be re -written as ???? = ( ?? 2 + sin ? ?? ) ???? Integrating, ? ? ???? = ? ? ( ?? 2 + sin ? ?? ) ???? ? ? ?? = ?? 3 3 - c os ? ?? + ?? Q7. The solution of the differential equation ?? ???? ???? = ?? ( ?????? ? ?? - ?????? ? ?? + ?? ) is (a) ?? = ?? ?? ???? (b) ?? + ?? ?? ???? = ?? (c) ?? + ?? ?? = ?? (d) None of these Ans: (a) Given equation may be expressed as ???? ???? = ?? ?? [ log ? ( ?? ?? ) + 1 ] Let ?? ?? = ?? ? ?? = ???? ? ???? ???? = ?? + ?? ???? ???? ? ? From (i), ?? + ?? ???? ???? = ?? ( log ? ?? + 1 ) ? ?? ???? ???? = ?? log ? ?? ? ???? ?? log ? ?? = ???? ?? ? ? ? 1 log ? ?? ?? ( log ? ?? ) = ? ? ???? ?? ? ? log ? ( log ? ?? ) = log ? ?? + log ? ?? ? log ? ( log ? ?? ) = log ? ( ???? ) ? log ? ?? = ???? ? ?? = ?? ???? ? ?? ?? = ?? ???? , ? ?? = ?? ?? ???? Q8. The general solution of the differential equation ( ?? + ?? ) ???? + ???? ?? = ?? is (a) ?? ?? + ?? ?? = ?? (b) ?? ?? ?? - ?? ?? = ?? (c) ?? ?? + ?? ???? = ?? (d) ?? ?? + ?? ???? = ?? Ans: (c) We have ???? ?? + ( ?? ?? ?? + ???? ?? ) = 0 ? ???? ?? + ?? ( ???? ) = 0 Integrating, ?? 2 2 + ???? = ?? 2 ? ?? 2 + 2 ???? = ?? Q9. Which of the following is a linear differential equation (a) ( ?? ?? ?? ?? ?? ?? ) ?? + ?? ?? ( ???? ???? ) ?? = ?? (b) ?? = ???? ???? + v ?? + ( ???? ???? ) ?? (c) ???? ???? + ?? ?? = ?????? ? ?? (d) ?? ???? ???? - ?? = ?? Ans: (c) (a), (b), (d) do not fulfill the criteria of a linear differential equation but (c) do. ???? ???? + ?? ?? = log ? ?? is a linear differential equation. Q10. Find the integral factor of equation ( ?? ?? + ?? ) ???? ???? + ?? ???? = ?? ?? - ?? Page 5 Solved Examples on Differential Equation JEE Main Single Correct Type Q1. The order and degree of the differential equation ?? = ?? ???? ???? + v ?? ?? ( ???? ???? ) ?? + ?? ?? are (a) 1,2 (b) 2,1 (c) 1,1 (d) 2,2 Ans: (a) Clearly, highest order derivative involved is ???? ???? , having order 1 . Expressing the above differential equation as a polynomial in derivative, we have ( ?? - ?? ???? ???? ) 2 = ?? 2 ( ???? ???? ) 2 + ?? 2 i.e., ( ?? 2 - ?? 2 ) ( ???? ???? ) 2 - 2 ???? ???? ???? + ?? 2 - ?? 2 = 0 In this equation, the power of highest order derivative is 2 . So its degree is 2 . Q2. The degree of the differential equation ?? ?? ?? ?? ?? ?? + ?? ( ???? ???? ) ?? = ?? ?? ?????? ? ( ?? ?? ?? ?? ?? ?? ) is (a) 1 (b) 2 (c) 3 (d) None of these Ans: (d) The above equation cannot be written as a polynomial in derivatives due to the term ?? 2 log ? ( ?? 2 ?? ?? ?? 2 ). Hence degree of the differential equation is 'not defined'. Q3. Differential equation whose general solution is ?? = ?? ?? ?? + ?? ?? ?? for all values of ?? ?? and ?? ?? is (a) ?? ?? ?? ?? ?? ?? + ?? ?? ?? + ???? ???? = ?? (b) ?? ?? ?? ?? ?? ?? + ?? ?? ?? - ???? ???? = ?? (c) ?? ?? ?? ?? ?? ?? - ?? ?? ?? ???? ???? = ?? (d) ?? ?? ?? ?? ?? ?? + ?? ?? ???? ???? - ?? ?? ?? = ?? Ans: (d) ?? = ?? 1 ?? + ?? 2 ?? (i) There are two arbitrary constants. To eliminate these constants, we need to differentiate (i) twice. Differentiating (i) with respect to ?? , ???? ???? = ?? 1 - ?? 2 ?? 2 Again differentiating with respect to ?? , ?? 2 ?? ?? ?? 2 = 2 ?? 2 ?? 3 From (iii), ?? 2 = ?? 3 2 ?? 2 ?? ?? ?? 2 and from (ii), ?? 1 = ???? ???? + ?? 2 ?? 2 ; ? ?? 1 = ???? ???? + ?? 2 ?? 2 ?? ?? ?? 2 From (i), ?? = ( ???? ???? + ?? 2 · ?? 2 ?? ?? ?? 2 ) ?? + ?? 2 2 · ?? 2 ?? ?? ?? 2 ? ?? = ?? 2 ?? 2 ?? ?? ?? 2 + ?? ???? ???? ? ?? 2 ?? ?? ?? 2 + 1 ?? ???? ???? - ?? ?? 2 = 0 Q4. ?? = ?? ?? + ?? is a solution of the differential equation (a) ?? ?? ???? ???? = ?? ?? (b) ?? ?? ???? ???? = ?? ?? (c) ?? ???? ???? = ?? (d) ?? ???? ???? = ?? Ans: (b) We have ?? = ?? ?? + 1 ? 1 ?? = ?? + 1 ?? = 1 + 1 ?? Differentiating w.r.t. ? - 1 ?? 2 ???? ???? = 0 - 1 ?? 2 ? ? ?? 2 ???? ???? = ?? 2 Q5. The differential equation of family of curves whose tangent form an angle of ?? / ?? with the hyperbola ???? = ?? ?? is (a) ???? ???? = ?? ?? + ?? ?? ?? ?? - ?? ?? (b) ???? ???? = ?? ?? - ?? ?? ?? ?? + ?? ?? (c) ???? ???? = - ?? ?? ?? ?? (d) None of these Ans: (b) The slope of the tangent to the family of curves is ?? 1 = ???? ???? Equation of the hyperbola is ???? = ?? 2 ? ?? = ?? 2 ?? ? ???? ???? = - ?? 2 ?? 2 ? Slope of tangent to ???? = ?? 2 is ?? 2 = - ?? 2 ?? 2 Now tan ? ?? 4 = ?? 1 - ?? 2 1 + ?? 1 ?? 2 ? 1 = ???? ???? + ?? 2 ?? 2 1 - ?? 2 ?? 2 ???? ???? ? ???? ???? ( 1 + ?? 2 ?? 2 ) = ( 1 - ?? 2 ?? 2 ) ? ???? ???? = ?? 2 - ?? 2 ?? 2 + ?? 2 Q5. The solution of the differential equation ( ?? + ?? ?? ) ???? ???? = ?? ( ?? + ?? ?? ) is (a) ?? ???? ?? - ?? ? ?? = ?????? ? ( ?? + ?? ?? ) + ?? (b) ???? ?? - ?? ? ?? = ?????? ? ( ?? + ?? ?? ) + ?? (c) ?? ???? ?? - ?? ? ?? + ?????? ? ( ?? + ?? ?? ) + ?? = ?? (d) None of these Ans: (a) Separating the variables, we can re-write the given differential equation as ???? ?? 1 + ?? 2 = ???? 1 + ?? 2 ? ? ? 2 ???? ?? 1 + ?? 2 = 2 ? ? ???? 1 + ?? 2 ? 2 tan - 1 ? ?? = log ?? ? ( 1 + ?? 2 ) + ?? Q6. The solution of ???? ???? = ?? ?? ?? + ?? ???? ? ?? is (a) ?? = ?? ?? ?? - ?? ???? ? ?? + ?? (b) ?? + ?? ???? ? ?? = ?? + ?? (c) ?? = ?? ?? ?? + ?? ???? ? ?? + ?? (d) None of these Ans: (a) Given equation may be re -written as ???? = ( ?? 2 + sin ? ?? ) ???? Integrating, ? ? ???? = ? ? ( ?? 2 + sin ? ?? ) ???? ? ? ?? = ?? 3 3 - c os ? ?? + ?? Q7. The solution of the differential equation ?? ???? ???? = ?? ( ?????? ? ?? - ?????? ? ?? + ?? ) is (a) ?? = ?? ?? ???? (b) ?? + ?? ?? ???? = ?? (c) ?? + ?? ?? = ?? (d) None of these Ans: (a) Given equation may be expressed as ???? ???? = ?? ?? [ log ? ( ?? ?? ) + 1 ] Let ?? ?? = ?? ? ?? = ???? ? ???? ???? = ?? + ?? ???? ???? ? ? From (i), ?? + ?? ???? ???? = ?? ( log ? ?? + 1 ) ? ?? ???? ???? = ?? log ? ?? ? ???? ?? log ? ?? = ???? ?? ? ? ? 1 log ? ?? ?? ( log ? ?? ) = ? ? ???? ?? ? ? log ? ( log ? ?? ) = log ? ?? + log ? ?? ? log ? ( log ? ?? ) = log ? ( ???? ) ? log ? ?? = ???? ? ?? = ?? ???? ? ?? ?? = ?? ???? , ? ?? = ?? ?? ???? Q8. The general solution of the differential equation ( ?? + ?? ) ???? + ???? ?? = ?? is (a) ?? ?? + ?? ?? = ?? (b) ?? ?? ?? - ?? ?? = ?? (c) ?? ?? + ?? ???? = ?? (d) ?? ?? + ?? ???? = ?? Ans: (c) We have ???? ?? + ( ?? ?? ?? + ???? ?? ) = 0 ? ???? ?? + ?? ( ???? ) = 0 Integrating, ?? 2 2 + ???? = ?? 2 ? ?? 2 + 2 ???? = ?? Q9. Which of the following is a linear differential equation (a) ( ?? ?? ?? ?? ?? ?? ) ?? + ?? ?? ( ???? ???? ) ?? = ?? (b) ?? = ???? ???? + v ?? + ( ???? ???? ) ?? (c) ???? ???? + ?? ?? = ?????? ? ?? (d) ?? ???? ???? - ?? = ?? Ans: (c) (a), (b), (d) do not fulfill the criteria of a linear differential equation but (c) do. ???? ???? + ?? ?? = log ? ?? is a linear differential equation. Q10. Find the integral factor of equation ( ?? ?? + ?? ) ???? ???? + ?? ???? = ?? ?? - ?? (a) ?? ?? + ?? (b) ?? ?? ?? ?? + ?? (c) ?? ?? - ?? ?? ?? + ?? (d) None of these Ans: (a) Given equation may be written as ???? ???? + 2 ?? ?? 2 + 1 ?? = ?? 2 - 1 ?? 2 + 1 Comparing with ???? ???? + ???? = ?? , ?? = 2 ?? ?? 2 + 1 I.F. = ?? ? ? ?? ???? = ?? ? ? 2 ?? ???? 1 + ?? 2 = ?? ln ? ( 1 + ?? 2 ) = 1 + ?? 2 Q11. The solution of the differential equation ( ?? + ?? ?? ) + ( ?? - ?? ?? ???? - ?? ? ?? ) ???? ???? = ?? is (a) ( ?? - ?? ) = ?? ?? ?? ???? - ?? ? ?? (b) ?? ?? ?? ?? ???? - ?? ? ?? = ?? ?? ?? ???? - ?? ? ?? + ?? (c) ?? ?? ?? ???? - ?? ? ?? = ???? ?? - ?? ? ?? + ?? (d) ?? ?? ?? ?? ???? - ?? ? ?? = ?? ?? ???? - ?? ? ?? + ?? Ans: (b) We have ( ?? - ?? tan - 1 ? ?? ) ???? ???? = - ( 1 + ?? 2 ) ? ???? ???? = - ( ?? - ?? t a n - 1 ? ?? 1 + ?? 2 ) ? ???? ???? + 1 1 + ?? 2 ?? = ?? t a n - 1 ? ?? 1 + ?? 2 This is a linear differential equation of the form ???? ???? + ?? ( ?? ) · ?? = ?? ( ?? ) ?? = 1 1 + ?? 2 , ? ?? = ?? tan - 1 ? ?? 1 + ?? 2 Integrating factor = ?? ? ? ?????? = ?? ? ? ???? 1 + ?? 2 = ?? tan - 1 ? ?? Multiplying (i) by I.F. and integrating, ?? ?? tan - 1 ? ?? = ? ?? t a n - 1 ? ?? 1 + ?? 2 · ?? tan - 1 ? ?? ???? = ? ( ?? t a n - 1 ? ?? ) 2 ???? 1 + ?? 2 = ( ?? t a n - 1 ? ?? ) 2 2 + ?? 2 ? 2 ?? ?? tan - 1 ? ?? = ?? 2 tan - 1 ? ?? + ?? Q12. Solution of ???? ???? - ?? ???? ?? ? ?? = - ?? ?? ?? ???? ? ?? is (a) ?? ?? ???? ? ?? = ???? ?? ? ?? + ?? (b) ?? ?? ?? ? ?? ?? = ???? ?? ? ?? + ?? (c) ?? ?? ???? ? ?? = ???? ?? ? ?? + ?? ? (d) None of theseRead More
About this Document
4.96/5 Rating
Apr 29, 2026 Last updated
Related Exams
Document Description: Differential Equation Solved Examples for JEE 2026 is part of Mathematics (Maths) for JEE Main & Advanced preparation. The notes and questions for Differential Equation Solved Examples have been prepared according to the JEE exam syllabus. Information about Differential Equation Solved Examples covers topics like and Differential Equation Solved Examples Example, for JEE 2026 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Differential Equation Solved Examples.
Introduction of Differential Equation Solved Examples in English is available as part of our Mathematics (Maths) for JEE Main & Advanced for JEE & Differential Equation Solved Examples in Hindi for Mathematics (Maths) for JEE Main & Advanced course. Download more important topics related with notes, lectures and mock test series for JEE Exam by signing up for free. JEE: Differential Equation Solved Examples
Description
Differential Equation Solved Examples of Maths covers all the important topics, helping you prepare for the JEE exam on EduRev. Start for free!
Information about Differential Equation Solved Examples
In this doc you can find the meaning of Differential Equation Solved Examples defined & explained in the simplest way possible. Besides explaining types of Differential Equation Solved Examples theory, EduRev gives you an ample number of questions to practice Differential Equation Solved Examples tests, examples and also practice JEE tests
Related Searches
pdf , past year papers, Viva Questions, Semester Notes, Extra Questions, video lectures, practice quizzes, Exam, study material, Previous Year Questions with Solutions, Differential Equation Solved Examples, Objective type Questions, ppt, Differential Equation Solved Examples, shortcuts and tricks, MCQs, Summary, Important questions, Differential Equation Solved Examples, mock tests for examination, Sample Paper, Free;