DIFFERENTIAL EQUATION

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1 MDE DIFFERENTIAL EQUATION NCERT Solved eamples upto the section 9. (Introduction) and 9. (Basic Concepts) : Eample : Find the order and degree, if defined, of each of the following differential equations : d (i) cos 0 (ii) 0 (iii) e 0 (i) order is one and degree is one (ii) order is two and degree is one (iii) order is three and degree is not defined. EXERCISE 9. Determine order and degree (if defined) of differential equations given in Eercises to 0. d ds d s. sin( ) s 0 dt dt d. cos 0 d 5. cos sin 5 6. ( ) ( ) () e 9. () 0 0. sin 0 d. The degree of the differential equation sin 0 is (a) (b) (c) (d) not defined d. The order of the differential equation 0 is (a) (b) (c) 0 (d) not defined. Order ; Degree not defined. Order ; Degree. Order ; Degree. Order ; Degree not defined 5. Order ; Degree 6. Order ; Degree 7. Order ; Degree 8. Order ; Degree 9. Order ; Degree 0. Order ; Degree. d. a NCERT Solved eamples upto the section 9. (General and Particular Solutions of a Differential Equation) : d Eample : Verif that the function = e is a solution of the differential equation 6 0. Eample : Verif that the function = a cos + b sin, where a, b R is a solution of the d differential equation 0. Einstein Classes, Unit No. 0, 0, Vardhman Ring Road Plaza, Vikas Puri Etn., Outer Ring Road New Delhi 0 08, Ph. : 96905, 857

2 EXERCISE 9. MDE In each of the Eercises to 0 verif that the given functions (eplicit or implicit) is a solution of the corresponding differential equation : In the following eercise verit that the given functions (eplicit or implicit) is a solution of the corresponding differential equations :. e : 0. = + + C : 0. = cos + C : sin 0. : 5. = A : ( 0) 6. = sin : ( 0 and > or < ) 7. = log + C : ( ) 8. cos = : ( sin + cos + ) 9. + = tan : 0 0. a ( a,a) : 0( 0). The number of arbitrar constants in the general solution of a differential equation of fourth order are (a) 0 (b) (c) (d). The number of arbitrar constants in the particular solution of a differential equation of third order are (a) (b) (c) (d) 0. d. d NCERT Solved eamples upto the section 9. (Formation of a Differential Equation whose General Solution is given) : Eample : Form the differential equation representing the famil of curves = m, where m is arbitrar constants. 0. Eample 5 : Form the differential equation representing the famil of curves = a sin ( + b), where a, b are arbitrar constants. d 0 Eample 6 : Form the differential equation representing the famil of ellipses having foci on -ais and centre at the origin. d 0 Einstein Classes, Unit No. 0, 0, Vardhman Ring Road Plaza, Vikas Puri Etn., Outer Ring Road New Delhi 0 08, Ph. : 96905, 857

3 MDE Eample 7 : Form the differential equation of the famil of circles touching the -ais at origin. Eample 8 : Form the differential equation representing the famil of parabolas having verte at origin and ais along positive direction of -ais. 0 EXERCISE 9. In each of the Eercises to 5, form a differential equation representing the given famil of curves b eliminating arbitrar constants a and b... = a (b ). = a e + b e a b. = e (a + b) 5. = e (a cos + b sin ) 6. Form the differential equation of the famil of circles touching the -ais at origin. 7. Form the differential equation of the famil of parabolas having verte at origin and ais along positive -ais. 8. Form the differential equation of the famil of ellipses having foci on -ais and centre at origin. 9. Form the differential equation of the famil of hperbolas having foci on -ais and centre at origin. 0. Form the differential equation of the famil of circles having centre on -ais and radius units.. Which of the following differential equations has = c e + c e as the general solution? d (a) 0 d d d (b) 0 (c) 0 (d) 0. Which of the following differential equations has = as one of its particular solution? (a) d d (b) d d (c) 0 (d) () ] () 0 9. () 0 0. ( 9) ( () 0. b. c NCERT Solved eamples upto the section 9.5 (Methods of Solving First Order, First Degree Differential Equations) : Eample 9 : Find the general solution of the differential equation,( ). Einstein Classes, Unit No. 0, 0, Vardhman Ring Road Plaza, Vikas Puri Etn., Outer Ring Road New Delhi 0 08, Ph. : 96905, 857

4 MDE Eample 0 : Find the general solution of the differential equation. Eample : Find the particular solution of the differential equation when = 0. given that =, Eample : Find the equation of the curve passing through the point (, ) whose differential equation is = ( + ) ( 0). = + log + c Eample : Find the equation of a curve passing through the point (, ), given that the slope of the tangent to the curve at an point (, ) is C. Eample : In a bank, principal increase continuousl at the rate of 5% per ear. In how man ears Rs 000 double itself? t = 0 log e EXERCISE 9. For each of the differential equations in Eercises to 0, find the general solution :. cos. ( ) cos. ( ). sec tan + sec tan = 0 5. (e + e ) (e e ) = 0 6. ( )( ) 7. log = sin 0. e tan + ( e ) sec = 0 For each of the differential equations in Eercises to, find a particular solution satisfing the given condition :. ( ) = + ; = when = 0. ( ) = ; = 0 when =. cos a (a R) ; = when = 0 Einstein Classes, Unit No. 0, 0, Vardhman Ring Road Plaza, Vikas Puri Etn., Outer Ring Road New Delhi 0 08, Ph. : 96905, 857

5 MDE 5. = tan ; = when = 0 5. Find the equation of a curve passing through the point (0, 0) and whose differential equation is e sin. 6. For the differential equation ( )( ), find the solution curve passing through the point (, ). 7. Find the equation of a curve passing through the point (0, ) given that at an point (, ) on the curve, the product of the slope of its tangent and -coordinate of the point is equal to the -coordinate of the point. 8. At an point (, ) of a curve, the slope of the tangent is twice the slope of the line segment joining the point of contact to the point (, ). Find the equation of the curve given that it passes through (, ). 9. The volume of spherical balloon being inflated changes at a constant rate. If initiall its radius is units and after seconds it is 6 units. Find the radius of balloon after t seconds. 0. In a bank, principal increases continuousl at the rate of r% per ear. Find the value of r if Rs 00 double itself in 0 ears (log e = 0.69).. In a bank, principal increases continuousl at the rate of 5% per ear. An amount of Rs 000 is deposited with this bank, how much will it worth after 0 ears (e 0.5 =.68).. In a culture, the bacteria count is,00,000. The number is increased b 0% in hours. In how man hours will the count reach,00,000, if the rate of growth of bacterial is proportional to the number present?. The general solution of the differential equation e is (a) e + e = C (b) e + e = C (c) e + e = C (d) e + e = C. tan C. = sin ( + C). = + Ae. tan tan = C 5. = log (e + e ) + C 6. tan C 7. = e c 8. + = C 9. sin C 0. tan = C( e ). log ( ) ( ) tan. log log. cos a. = sec 5. = e (sin cos ) 6. + = log ( ) + ) ) 7. = 8. ( + ) = + 9. (6t + 7) / 0. (6.9). Rs. 68. log log 0. a Einstein Classes, Unit No. 0, 0, Vardhman Ring Road Plaza, Vikas Puri Etn., Outer Ring Road New Delhi 0 08, Ph. : 96905, 857

6 MDE 6 Eample 5 : Show that the differential equation ( ) is homogeneous and solve it. Eample 6 : Show that the differential equation solve it. cos cos is homogeneous and Eample 7 : Show that the differential equation e e 0 is homogeneous and find its particular solution, given that, = 0 when =. Eample 8 : Show that the famil of curves for which the slope of the tangent at an point (, ) on it is, is given b = c. EXERCISE 9.5 In each of the Eercise to 0, show that the given differential equation homogeneous and solve each of them.. ( + ) = ( + ).. ( ) ( + ) = 0. ( ) + = cos sin sin cos 8. sin 0 9. log 0 0. e e 0 For each of the differential equations in Eercises from to 5, find the particular solution satisfing the given conditions :. ( + ) + ( ) = 0; = when =. + ( + ) = 0; = when =. sin 0; when. cosec 0; 0 when 5. 0; when Einstein Classes, Unit No. 0, 0, Vardhman Ring Road Plaza, Vikas Puri Etn., Outer Ring Road New Delhi 0 08, Ph. : 96905, 857

7 MDE 7 6. A homogeneous differential equation of the from h can be solved b making the substitution. (a) = v (b) v = (c) = v (d) = v 7. Which of the following is a homogeneous differential equation? (a) ( ) ( + + ) = 0 (b) () ( + ) = 0 (c) ( + ) + = 0 (d) + ( ) = 0. ( ) Ce. = log + C. tan log( ) C 5. log log C. + = C 6. C 7. cos C 8. cos Csin e 9. c log 0. C. log( ) tan log. + =. cot log e. cos log e 6. c 7. d 5. ( 0, e) log Eample 9 : Find the general solution of the differential equation cos. sin cos Ce ] Eample 0 : Find the general solution of the differential equation C ( 0) Eample : Find the general solution of the differential equation ( + ) = 0. = + C Einstein Classes, Unit No. 0, 0, Vardhman Ring Road Plaza, Vikas Puri Etn., Outer Ring Road New Delhi 0 08, Ph. : 96905, 857

8 MDE 8 Eample : Find the particular solution of the differential equation cot cot ( 0) given that = 0 when. (sin 0) sin Eample : Find the equation of a curve passing through the point (0, ). If the slope of the tangent to the curve at an point (, ) is equal to the sum of the coordinate (abscissa) and the product of the coordinates and coordinates (ordinate) of that point. e EXERCISE 9.6 For each of the differential equations in Eercises to, find the general solution :. sin. e.. (sec ) tan 0 5. cos tan 0 6. log 7. log log 8. ( + ) + = cot ( 0) 9. cot 0( 0) 0. ( ). + ( ) = 0. ( ) ( 0) For each of the differential equations given in Eercises to 5, find a particular solution satisfing the given condition :. tan sin ; 0 when. ( ) ; 0 when 5. cot sin ; when 6. Find the equation of a curve passing through the origin given that the slope of the tangent to the curve at an point (, ) is equal to the sum of the coordinates of the point. 7. Find the equation of a curve passing through the point (0, ) given that the sum of the coordinates of an point on the curve eceeds the magnitude of the slope of the tangent to the curve at that point b The Integrating Factor of the differential equation is (a) e (b) e (c) (d) Einstein Classes, Unit No. 0, 0, Vardhman Ring Road Plaza, Vikas Puri Etn., Outer Ring Road New Delhi 0 08, Ph. : 96905, 857

9 MDE 9 9. The Integrating Factor of the differential equation ( ) a( ) is (a) (b) (c) (d). (sin cos ) Ce. = e + Ce. C 5. (sec + tan ) = sec + tan + C 5. = (tan ) + Ce tan 6. (log ) C 7. log ( log ) C 6 8. = ( + ) log sin + C( + ) 9. cot C sin 0. ( + + ) = C e.. = + C. = cos cos. ( + ) = tan 5. = sin sin = e 7. = e 8. c 9. d C MISCELLANEOUS EXAMPLES : Eample : Verif that the function = c e a cos b + c e a sin b, where c, c are arbitrar d constants is a solution of the differential equation a (a b ) 0. Eample 5 : Form the differential equation of the famil of circles in the second quadrant and touching the coordinates aes. ( ) [() ] [ ] Eample 6 : Find the particular solution of the differential equation = 0 when = 0. e + e 7 = 0 log given that Eample 7 : Solve the differential equation ( ) sin ( ) cos. sec C Eample 8 : Solve the differential equation (tan ) = ( + ). (tan ) Ce tan Einstein Classes, Unit No. 0, 0, Vardhman Ring Road Plaza, Vikas Puri Etn., Outer Ring Road New Delhi 0 08, Ph. : 96905, 857

10 MISCELLANEOUS EXERCISE ON CHAPTER 9. For each of the differential equations given below, indicate its order and degree (if defined). MDE 0 (i) d 5 6 log (ii) 7 sin d d (iii) sin 0. For each of the eercises given below, verif that the given function (implicit or eplicit) is a solution of the corresponding differential equation. d (i) = a e + b e + : 0 d (ii) = e (a cos + b sin ) : 0 d (iii) = sin : 9 6cos 0 (iv) = log : ( ) 0. Form the differential equation representing the famil of curves given b ( a) + = a, where a is an arbitrar constant.. Prove that = c ( + ) is the general solution of differential equation ( ) = ( ), where c is a parameter. 5. Form the differential equation of the famil of circles in the first quadrant which touch the coordinate aes. 6. Find the general solution of the differential equation Show that the general solution of the differential equation 0 is given b ( + + ) = A ( ), where A is parameter. 8. Find the equation of the curve passing through the point 0, whose differential equation in sin cos + cos sin = Find the particular solution of the differential equation ( + e ) + ( + ) e = 0, given that = when = Solve the differential equation e e ( 0).. Find a particular solution of the differential equation ( ) ( + ) =, given that =, when = 0. (Hint : put = t). e. Solve the differential equation ( 0). Einstein Classes, Unit No. 0, 0, Vardhman Ring Road Plaza, Vikas Puri Etn., Outer Ring Road New Delhi 0 08, Ph. : 96905, 857

11 MDE. Find a particular solution of the differential equation cot cosec ( 0), given that = 0 when.. Find a particular solution of the differential equation ( + ) = e, given that = 0 when = The population of a village increases continuousl at the rate proportional to the number of its inhabitants present at an time. If the population of the village was 0,000 in 999 and 5000 in the ear 00, what will be the population of the village in 009? 6. The general solution of the differential equation 0 is (a) = C (b) = C (c) = C (d) = C 7. The general solution of a differential equation of the tpe P Q is (a) P P e Qe C (b) P P.e Qe C (c) P P e Qe C (d) P P e Qe C 8. The general solution of the differential equation e + ( e + ) = 0 is (a) e + = C (b) e + = C (c) e + = C (d) e + = C. (i) Order ; Degree (ii) Order ; Degree (iii) Order ; Degree not defined. 5. ( ) ( ) ( () ) 6. sin + sin = C 8. cos sec 9. tan tan (e ) e 0. C. log = + +. e ( C). sin (sin 0). log, c 7. c 8. c Einstein Classes, Unit No. 0, 0, Vardhman Ring Road Plaza, Vikas Puri Etn., Outer Ring Road New Delhi 0 08, Ph. : 96905, 857

Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi , Ph. : ,

Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi , Ph. : , C1 Order and Degree of a Differential Equation : Order : Order is the highest differential appearing in a differential equation. MDE 1 DIFFERENTIAL EQUATION Degree : It is determined b the degree of the