First Order Differential Equations

These are questions on first order differential equations

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1. Created by T. Madas st 1 ORDER O.D.E. EXAM QUESTIONS Created by T. Madas
2. Created by T. Madas Question 1 (**) dy 4 y + = 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 ) . FP2-Q , y = x 2 − x + Created by T. Madas 1 x4
3. Created by T. Madas Question 2 (**+) dy + y tan x = e2 x cos x , y ( 0 ) = 2 . dx Show that the solution of the above differential equation is y= 1 2x e + 3 cos x . 2 ( ) FP2-R , proof Created by T. Madas
4. Created by T. Madas Question 3 (**+) The velocity of a particle v ms −1 at time t s satisfies the differential equation t dv = v +t , t > 0. dt Given that when t = 2 , v = 8 , show that when t = 8 v = 16 ( 2 + ln 2 ) . proof Question 4 (**+) x dy + 4 y = 8 x 4 , subject to y = 1 at x = 1 . dx Show that the solution of the above differential equation is y = x4 . proof Created by T. Madas
5. Created by T. Madas Question 5 (***) dy sin x = sin x sin 2 x + y cos x . dx Given that y = 3 π π at x = , find the exact value of y at x = . 2 6 4 1+ 2 Question 6 (***) 1 dy 3 x + 2 y = 9 x x + 1 2 , with y = 27 at x = 2 . 2 dx ( ) Show that the solution of the above differential equation is y= 2 x2 ( ) 3 x3 + 1 2 . proof Created by T. Madas
6. Created by T. Madas Question 7 (***) A trigonometric curve C satisfies the differential equation dy cos x + y sin x = cos3 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 ≤ 2π . The sketch must show clearly the coordinates of the points where the graph of C meets the x axis. y = sin x cos x + A cos x Created by T. Madas
7. Created by T. Madas Question 8 (***) 20 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 2M + +1 = 0 , t ≥ 0 . dt 20 − t Show clearly that M = 1 (10 − t )( 20 − t ) . 10 proof Created by T. Madas
8. Created by T. Madas Question 9 (***+) Given that z = f ( x ) and y = g ( x ) satisfy the following differential equations dz + 2 z = e −2 x and dx dy + 2y = z , dx a) Find z in the form z = f ( x ) b) Express y in the form y = g ( x ) , given further that at x = 0 , y = 1 , ( dy =0 dx ) z = ( x + C ) e −2 x , y = 1 x 2 + 2 x + 1 e−2 x 2 Created by T. Madas
9. Created by T. Madas Question 10 (***+) x dy = dx y 2 + 1 , x > 0 , with y = 0 at x = 2 . Show that the solution of the above differential equation is y= x 1 − . 4 x proof Question 11 (***+) ( x + 1) dy = y + x + x 2 , x > −1 . dx Given that y = 2 at x = 1 , solve the above differential equation to show that y = 4 ( 3 − ln 2 ) at x = 3 . proof Created by T. Madas
10. Created by T. Madas Question 12 (***+) dy + ky = cos3 x , 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. y = A e− x + Created by T. Madas k 9+k 2 cos3x + 3 9 + k2 sin 3x
11. Created by T. Madas Question 13 (***+) ( 2 x − 4 y 2 ) dydx + y = 0 . By reversing the role of x and y in the above differential equation, or otherwise, find its general solution. FP2-L , xy 2 = y 4 + C Created by T. Madas
12. Created by T. Madas Question 14 (****) The curve with equation y = f ( x ) satisfies x dy + (1 − 2 x ) y = 4 x , x > 0 , f (1) = 3 e 2 − 1 . dx ( ) Determine an equation for y = f ( x ) . 3 1 y = ex − − 2 x x Created by T. Madas
13. Created by T. Madas Question 15 (****) A curve C , with equation y = f ( x ) , passes through the points with coordinates (1,1) and ( 2, k ) , where k is a constant. Given further that the equation of C satisfies the differential equation x2 dy + xy ( x + 3) = 1 , dx determine the exact value of k . FP2-J , k = Created by T. Madas e +1 8e
14. Created by T. Madas Question 16 (****) dy + y = (1 − x 2 ) (1 − x ) , −1 < x < 1 . (1 − x2 ) dx 1 2 2 1 at x = , show that the solution of the above differential equation 2 2 can be written as Given that y = y= 2 3 (1 − x2 ) (1 + x ) . FP2-O , proof Created by T. Madas
15. Created by T. Madas 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 dy = x − 2y . dx a) Determine an equation of C . b) Sketch the graph of C . 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 − 1 + 5 e−2 x 2 4 4 Created by T. Madas
16. Created by T. Madas Question 18 (****) dy y 5 + = , x >0. dx x x2 + 2 4 x2 + 3 ( )( ) Given that y = 1 ln 7 at x = 1 , show that the solution of the above differential equation 2 6 can be written as 1  4 x2 + 3  ln  y= . 2 x  2 x 2 + 4  FP2-M , proof Created by T. Madas
17. Created by T. Madas Question 19 (****) x 2 dy + 3 y = x e− x , x > 0 . dx Show clearly that the general solution of the above differential equation can be written in the form ( ) 2 2 yx3 + x 2 + 1 e− x = constant . proof Created by T. Madas
18. Created by T. Madas Question 20 (****) The general point P lies on the curve with equation y = f ( x ) . The gradient of the curve at P is 2 more than the gradient of the straight line segment OP . Given further that the curve passes through Q (1, 2 ) , express y in terms of x . y = 2 x (1 + ln x ) Created by T. Madas
19. Created by T. Madas Question 21 (****+) A curve with equation y = f ( x ) passes through the origin and satisfies the differential equation ( 2 y 1+ x 2 ) 3 dy 2 2 2 + xy = 1 + x . dx ( ) By finding a suitable integrating factor, or otherwise, show clearly that 2 y = x3 + 3x 3 x2 + 1 . SPX-V , proof Created by T. Madas
20. Created by T. Madas Question 22 (****+) The curve with equation y = f ( x ) passes through the origin, and satisfies the relationship d  y x 2 + 1  = x5 + 2 x3 + x + 3 xy .  dx  ( ) Determine a simplified expression for the equation of the curve. 1 2 SPX-I , y = 1 x 2 + 1 − 1 x 2 + 1 2 3 3 ( Created by T. Madas ) ( )
21. Created by T. Madas Question 23 (****+) A curve with equation y = f ( x ) passes through the point with coordinates ( 0,1) and satisfies the differential equation y2 dy + y 3 = 4e x . dx By finding a suitable integrating factor, solve the differential equation to show that y 3 = 3e x − 2e −3 x . SPX-E , proof Created by T. Madas
22. Created by T. Madas Question 24 (****+) π π  It is given that a curve with equation y = f ( x ) passes through the point  ,  and 4 4 satisfies the differential equation  dy   − tan x  sin 2 x = y .  dx  Find an equation for the curve in the form y = f ( x ) . SPX-H , y = x tan x Created by T. Madas
23. Created by T. Madas Question 25 The variables x and y satisfy (2 y − x) If y = 1 at x = 2 , show that x = y + dy = y , y > 0, x > 0 . dx 1 . y V , FP3-T , proof Created by T. Madas
24. Created by T. Madas Question 26 The variables x and y satisfy y ( y + 1) dy = , y > 0. dx y − x − xy − 1 If y = 1 at x = 1 − ln 4 , show that y + ln ( y + 1) = 0 at x = 3 . V , SPX-O , proof Created by T. Madas
25. Created by T. Madas Question 27 (*****) The curve with equation y = f ( x ) has the line y = 1 as an asymptote and satisfies the differential equation x3 dy − x = xy + 1 , x ≠ 0 . dx Solve the above differential equation, giving the solution in the form y = f ( x ) . −1 SPX-J , y = e x − Created by T. Madas 1 x
26. Created by T. Madas Question 28 (*****) ( ) It is given that a curve with equation x = f ( y ) passes through the point 0, 1 and 2 satisfies the differential equation ( 2 y + 3x ) dy = y. dx Find an equation for the curve in the form x = f ( y ) . SPX-D , x = 4 y 3 − y Created by T. Madas
27. Created by T. Madas Question 29 (*****) Use suitable manipulations to solve this exact differential equation. 4x dy + sin 2 y = 4 cos 2 y , dx ( ) y 1 =0. 4 Given the answer in the form y = f ( x ) . 1   SPX-P , y = arctan  2 − x   Created by T. Madas