1 st ORDER O.D.E. EXAM QUESTIONS

Transcription

1 1 st ORDER O.D.E. EXAM QUESTIONS

2 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 in the form y = f ( x). 1 4 y = x x + x Question (**+) x y tan x e cos x dx + =, with y = at x = 0. Show that the solution of the above differential equation is x ( ) 1 y = e + 3 cos x. proof

3 Question 3 (**+) The velocity of a particle v 1 ms at time t s satisfies the differential equation dv t v t dt = +, t > 0. Given that when t =, v = 8, show that when t = 8 v = 16( + ln ). proof Question 4 (**+) 4 x + 4y = 8x, subject to 1 dx y = at x = 1. Show that the solution of the above differential equation is 4 y = x. proof

4 Question 5 (***) sin x = sin x sin x + y cos x. dx Given that 3 y = at π x =, find the exact value of y at 6 π x = Question 6 (***) 3 ( ) 1 x y 9x x 1 dx + = +, with y = 7 at x =. Show that the solution of the above differential equation is 3 ( ) 3 x 1 y = +. x proof

5 Question 7 (***) A trigonometric curve C satisfies the differential equation 3 cos x + ysin x = cos x. dx a) Find a general solution of the above differential equation. b) Given further that the curve passes through the Cartesian origin O, sketch the graph of C for 0 x π. The sketch must show clearly the coordinates of the points where the graph of C meets the x axis. y = sin xcos x + Acos x

6 Question 8 (***) 0 grams of salt are dissolved into a beaker containing 1 litre of a certain chemical. The mass of salt, M grams, which remains undissolved t seconds later, is modelled by the differential equation dm dt M = 0 0 t, t 0. Show clearly that M = 1 ( 10 t)( 0 t 10 ). proof

7 Question 9 (***+) Given that z = f ( x) and y g ( x) = satisfy the following differential equations a) Find z in the form z = f ( x) b) Express y in the form y g ( x) dz z e x + = and y z dx dx + =, =, given further that at x = 0, y = 1, 0 dx = = ( + ) e x, y = ( x + x + ) z x C 1 1 e x

8 Question 10 (***+) x y 1 dx = +, x > 0, with y = 0 at x =. Show that the solution of the above differential equation is x 1 y =. 4 x proof Question 11 (***+) ( x ) y x x + 1 = + +, x > 1. dx Given that y = at x = 1, solve the above differential equation to show that y = 4( 3 ln ) at x = 3. proof

9 Question 1 (***+) + ky = cos3x, k is a non zero constant. dx By finding a complimentary function and a particular integral, or otherwise, find the general of the above differential equation. x k 3 y = A e + cos3x + sin 3x 9 + k 9 + k

10 Question 13 (***+) ( ) x 4y + y = 0. dx By reversing the role of x and y in the above differential equation, or otherwise, find its general solution. 4 xy = y + C

11 Question 14 (****) The curve with equation y = f ( x) satisfies x ( 1 x) y 4x dx + =, 0 Determine an equation for y = f ( x). x >, ( ) 1 3( e 1) f =. 3 x 1 y = e x x

12 Question 15 (****) A curve C, with equation y = f ( x), passes through the points with coordinates ( 1,1 ) and (,k ), where k is a constant. Given further that the equation of C satisfies the differential equation determine the exact value of k. x xy( x 3) 1 dx + + =, e + 1 k = 8e

13 Question 16 (****) ( 1 x ) y ( 1 x )( 1 x) 1 + =, 1< x < 1. dx Given that y = at can be written as 1 x =, show that the solution of the above differential equation ( 1 )( 1 ) y = x + x. 3 FP-O, proof

14 Question 17 (****) A curve C, with equation y = f ( x), meets the y axis the point with coordinates ( 0,1 ). It is further given that the equation of C satisfies the differential equation a) Determine an equation of C. b) Sketch the graph of C. x y dx =. The graph must include in exact simplified form the coordinates of the stationary point of the curve and the equation of its asymptote. SYNF-A, y = 1 x e x

15 Question 18 (****) y 5 + = dx x x x ( + )( 4 + 3), x > 0. Given that y = 1 ln 7 at x = 1, show that the solution of the above differential equation 6 can be written as 1 4x + 3 y = ln x. x + 4 FP-M, proof

16 Question 19 (****) x + 3y = x e x, x > 0. dx Show clearly that the general solution of the above differential equation can be written in the form ( x ) 3 x yx e = constant. proof

17 Question 0 (****) The general point P lies on the curve with equation y = f ( x). The gradient of the curve at P is more than the gradient of the straight line segment OP. Given further that the curve passes through Q ( 1, ), express y in terms of x. ( ) y = x 1+ ln x

18 Question 1 (****+) A curve with equation y = f ( x) passes through the origin and satisfies the differential equation ( ) ( ) 3 y 1+ x + xy = 1+ x. dx By finding a suitable integrating factor, or otherwise, show clearly that y 3 x + 3x =. 3 x + 1 SPX-V, proof

19 Question (***+) The curve with equation y = f ( x) passes through the origin, and satisfies the relationship ( ) d 5 3 y x + 1 = x + x + x + 3xy. dx Determine a simplified expression for the equation of the curve. y = 1 x x SPX-I, ( ) ( ) 1

20 Question 3 (****+) A curve with equation y = f ( x) passes through the point with coordinates ( 0,1 ) and satisfies the differential equation y y 4e x dx + =. 3 By finding a suitable integrating factor, solve the differential equation to show that y 3 x 3x = 3e e. SPX-E, proof

21 Question 4 (****+) It is given that a curve with equation y f ( x) satisfies the differential equation π π = passes through the point, 4 4 and tan x sin x = y. dx Find an equation for the curve in the form y = f ( x). SPX-H, y = x tan x

22 Question 5 (*****) The curve with equation y = f ( x) has the line y = 1 as an asymptote and satisfies the differential equation 3 x x xy 1 dx = +, x 0. Solve the above differential equation, giving the solution in the form y = f ( x). SPX-J, 1 1 e x y = x

23 Question 6 (*****) It is given that a curve with equation x f ( y) satisfies the differential equation ( 3 ) = passes through the point 1 ( ) y + x = y. dx Find an equation for the curve in the form x = f ( y). 0, and SPX-D, 3 x = 4y y