2 nd ORDER O.D.E.s SUBSTITUTIONS

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1 nd ORDER O.D.E.s SUBSTITUTIONS

2 Question 1 (***+) d y y 8y + 16y = d d d, y 0, Find the general solution of the above differential equation by using the transformation equation t = y. Give the answer in the form y = f ( ). ( e e ) y = A + B

3 Question (***+) The differential equation d y 3, d + d =, 0 is to be solved subject to the boundary conditions 3 y =, 1 d = at = 1. a) Show that the substitution into v =, transforms the above differential equation d dv v + = 3. d b) Hence find the solution of the original differential equation, giving the answer y = f. in the form ( ) y = + +

4 Question 3 (***+) The curve C has equation y = f ( ) and satisfies the differential equation ( ) d y 3 y 1 3 e d d =, 0 is to be solved subject to the boundary conditions 3 y =, 1 d = at = 1. a) Show that the substitution y = v, where v is a function of transforms the above differential equation into d v 4v 3e d =. It is further given that C meets the ais at = ln and has a finite value for y as gets infinitely negatively large. b) Epress the equation of C in the form y = f ( ). 1 e e y =

5 Question 4 (****) The differential equation ( ) 3 d y =, d d is to be solved subject to the boundary conditions y = 0, 4 d = at = 0. Use the substitution u =, where u is a function of, to show that the solution d of the above differential equation is 4 y = FP3-O, proof

6 Question 5 (****) d y 6 9y 7 6y ( ) d + + d + =. Use the substitution u = y, where u is a function of, to find a general solution of the above differential equation. A y = e + B e

7 Question 6 (****) By using the substitution z =, or otherwise, solve the differential equation d ( ) d y d + d = +, subject to the conditions = 0, y =, 1 d = y = + + arctan

8 Question 7 (****) d y ( 1 6e ) 10y e 5e sin( e ) d d + =. a) By using the substitution = ln t or otherwise, show that the above differential equation can be transformed to d y y = 5sin t. dt dt b) Hence find a general solution for the original differential equation. 3e FP3-M, 1 1 y = e Acos( e ) + Bsin( e ) + sin ( e ) cos( e ) 6 3

9 Question 8 (***+) Given that if = e t and y f ( ) =, show clearly that a) d =. dt b) d y d y d = dt dt. The following differential equation is to be solved subject to the boundary conditions d y d d + = 3 4y ln 1 y =, 3 d = at = 1. c) Use the substitution e t = to solve the above differential equation. ( ) y = ln

10 Question 9 (****) d y 4 tan ysec 0 d d =. The above differential equation is to be solved by a substitution. a) If t = tan show that i. ii. = sec d dt d y 4 = d y sec + sec tan d dt dt b) Use the results obtained in part (a) to find a general solution of the differential y = f. equation in the form ( ) FP3-Q, tan y = Ae + Be tan

11 Question 10 (****) Show clearly that the substitution z = sin, transforms the differential equation d y 3 5 cos + sin ycos = cos, d d into the differential equation ( z ) d y y = 1 dz proof

12 Question 11 (****+) Use the substitution z = y, where y = f ( ), to solve the differential equation d y y 0 + =, d y d d subject to the boundary conditions y = 4, 44 d = at = 0. Give the answer in the form y = f ( ). 6 4 y = 9e 6e + e

13 Question 1 (****+) d y y 0 + =. d d The above differential equation is to be solved by a substitution. a) Given that y = f ( ) and 1 t =, show clearly that i. ii. 1 =. d t dt d y 1 d y 1 3 d = 4t dt 4t dt. b) Hence show further that the differential equation d y y 0 + =, d d can be transformed to the differential equation d y 3 + y = 0. dt dt c) Find a general solution of the original differential equation, giving the answer y = f. in the form ( ) y = Ae + Be

14 Question 13 (****+) Show clearly that the substitution equation z = y, where y f ( ) =, transforms the differential d y y 0 + =, d y d d into the differential equation d z dz 5 4z 0 d d + = proof

15 Question 14 (****+) Given that if =, where y f ( ) t 1 =, show clearly that a) b) 1 = t. d dt d y d y 4t d = dt + dt. The following differential equation is to be solved subject to the boundary conditions ( ) d y d + d + =, y 10 y =, 3 c) Show further that the substitution d y 10 =. d = at 0 =, where y f ( ) t 1 above differential equation into the differential equation =, transforms the d y 4 + 3y = 3t. dt dt d) Show that a solution of the original differential equation is 3 4 y = e + e proof

16 Question 15 (****+) The curve with equation y = f ( ) satisfies d y 5 13y 0 d + d + =, > 0. By using the substitution given further that y = 1 and d = at = 1. = e t, or otherwise, determine an equation for y f ( ) =, ( ) cos 3ln y =

17 Question 16 (****+) d y 3 cot y cosec cos cos d d + =. Use the substitution y = z sin, where z is a function of, to solve the above differential equation subject to the boundary conditions y = 1, 0 d = at π =. Give the answer in the form ( ) y = a sin + b 1 sin sin, where a and b are constants to be found. a = 1, b = 1 3

18 Question 17 (****+) The function y = f ( ) satisfies the following relationship. d y d d = 0. d It is further given that = t and y = ln v. Show that d v = v. dt proof

19 Question 18 (****+) d y d d + =. y Use the substitution differential equation. =, where y f ( ) z 1 =, to find a general solution of the above 1 1 y = Ae + B e +

20 Question 19 (*****) Use a suitable substitution to solve the differential equation d y 6 ln 6 ( ln ) d =, y 1 = 3 d subject to the boundary conditions y ( 1) = 1, ( ) Give a simplified answer in the form y = f ( ). 3 FP3-S, ( ln ) y = +

21 Question 0 (*****) The function with equation y = f ( ) satisfies the differential equation d y 1 y ln3 = d y d d, y ( 0) = 1, ( ) 0 = ln3. Solve the above differential equation by using the substitution p =, to show that d 3 + y =. SPX-G, proof

22 Question 1 (*****) d y d d = 1. d By using the substitution t =, or otherwise, show that the general solution of the above differential equation is y = A + ln 1+ Be, where A and B are arbitrary constants. SPX-C, proof

1 st ORDER O.D.E. EXAM QUESTIONS

1 st ORDER O.D.E. EXAM QUESTIONS Question 1 (**) 4y + = 6x 5, x > 0. dx x Determine the solution of the above differential equation subject to the boundary condition is y = 1 at x = 1. Give the answer