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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
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