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Differential Equations and Modeling · CAPE Pure Mathematics Unit 2

69 past-paper questions on Differential Equations and Modeling, part of Counting, Matrices and Differential Equations, from every CAPE Pure Mathematics Unit 2 paper on Quelpr.

Practise Differential Equations and Modeling
  1. 2(a)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (Rest of Region) · Paper 2Solve the differential equation \frac{\mathrm{d}y}{\mathrm{d}x} + y = e^{2x}.
  2. 2(a)(ii)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (T&T) · Paper 2Solve the differential equation x^2 \frac{\mathrm{d}y}{\mathrm{d}x} + xy = \ln x.
  3. 2(b)(i)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (T&T) · Paper 2Find the values of the constants m and n, given that y = m \cos x + n \sin x satisfies the differential equation \frac{\mathrm{d}^2y}{\mathrm{d}x^2} - 4\frac{\mathrm{d}y}{\mathrm{d}x} + 3y = 10 \sin x.
  4. 2(b)(ii)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (T&T) · Paper 2Hence, find the general solution of the differential equation.
  5. 1(a)(iii)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2010 · Paper 2Determine the time when the temperature of the water in the tank is 70^\circ\text{ C}.
  6. 2(b)(i)7 marks· CAPE Pure Mathematics Unit 2 · May/June 2010 · Paper 2Using a suitable integrating factor, show that the general solution of this differential equation is y = t + 10 + \frac{c}{(t+10)^2}, where c is an arbitrary constant.
  7. 2(b)(ii)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2010 · Paper 2Given that the tank initially contains 5\text{ kg} of salt in the liquid, calculate the amount of salt that dissolves in the tank of water at t = 15.
  8. 6(a)(i)7 marks· CAPE Pure Mathematics Unit 2 · May/June 2013 · Paper 2Obtain the general solution of the differential equation \cos x \frac{dy}{dx} + y\sin x = 2x\cos^2 x.
  9. 6(a)(ii)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2013 · Paper 2Hence, given that y = \frac{15\sqrt{2}\pi^2}{32} when x = \frac{\pi}{4}, determine the constant of the integration.
  10. 6(b)(i)a)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2013 · Paper 2The general solution of the differential equation y'' + 2y' + 5y = 4\sin 2t is y = CF + PI, where CF is the complementary function and PI is a particular integral. Calculate the roots of…
  11. 6(b)(i)b)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2013 · Paper 2Hence, obtain the complementary function (CF), the general solution of y'' + 2y' + 5y = 0.
  12. 6(b)(ii)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2013 · Paper 2Given that the form of the particular integral (PI) is u_p(t) = A\cos 2t + B\sin 2t, show that A = -\frac{16}{17} and B = \frac{4}{17}.
  13. 6(b)(iii)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2013 · Paper 2Given that y(0) = 0.04 and y'(0) = 0, obtain the general solution of the differential equation.
  14. 6(a)(i)10 marks· CAPE Pure Mathematics Unit 2 · May/June 2014 · Paper 2Show that the general solution of the differential equation y' + y\tan x = \sec x is y = \sin x + C\cos x.
  15. 6(a)(ii)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2014 · Paper 2Hence, obtain the particular solution where y = \frac{2}{\sqrt{2}} and x = \frac{\pi}{4}.
  16. 6(b)11 marks· CAPE Pure Mathematics Unit 2 · May/June 2014 · Paper 2A differential equation is given as y'' - 5y' = x e^{5x}. Given that a particular solution is y_p(x) = Ax^2 e^{5x} + Bx e^{5x}, solve the differential equation.
  17. Q421 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 1If the auxiliary equation for a homogeneous second order differential equation with real, constant coefficients is given by \lambda^2 + 6\lambda + 50 = 0, then the general solution of the differential equation may be…
  18. Q431 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 1The general solution of the differential equation (x - 2)\frac{dy}{dx} = y is
  19. Q441 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 1Given that y = \frac{\pi}{4} and x = \frac{1}{2}, then the particular solution of \frac{dy}{dx} = 2x\cos^2 y is
  20. Q451 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 1The general solution of the differential equation \frac{dy}{dx} = \frac{y}{x} is
  21. 6(b)(i)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 2Show that the equation y + xy + x^2 = 0 is a solution of the differential equation \frac{dy}{dx} = \frac{y - x^2}{x(1 + x)}.
  22. 6(b)(ii)a)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 2Find the general solution of the differential equation.
  23. 6(b)(ii)b)7 marks· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 2Hence, show that the solution which satisfies the boundary conditions y(0) = 1 and y'\left(\frac{\sqrt{2}}{2}\right) = 0 is y = \frac{1}{e^2 + 1}\left(e^{\sqrt{2}x} + e^{2 - \sqrt{2}x}\right).
  24. Q341 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 1Given that y = 0 at x = 0, the general solution of the differential equation y'' + 6y' + 9y = 0 is
  25. Q411 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 1The general solution of the differential equation \frac{dy}{dx} = \frac{y}{x} is
  26. Q421 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 1A suitable integrating factor for the solution of the differential equation \frac{dy}{dx} + \frac{2y}{x} = \frac{1}{x} is
  27. 6(b)6 marks· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 2Determine whether y = C_1 x + C_2 x^2 is a solution to the differential equation (x^2 / 2) y'' - x y' + y = 0, where C_1 and C_2 are constants.
  28. 6(c)(i)7 marks· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 2Show that the general solution to the differential equation 3(x^2 + x) dy/dx = 2y(1 + 2x) is y = C * ((x^2 + x)^2)^(1/3), where C in R.
  29. 6(c)(ii)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 2Hence, given that y(1) = 1, solve 3(x^2 + x) dy/dx = 2y(1 + 2x).
  30. Q421 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 1If the auxiliary equation for a homogeneous second order differential equation with real, constant coefficients is given by \lambda^2 + 6\lambda + 50 = 0, then the general solution of the differential equation may be…
  31. Q431 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 1A suitable integrating factor for the solution of the differential equation \frac{dy}{dx} + \frac{2y}{x} = \frac{1}{x} is
  32. Q441 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 1Given that y = \frac{\pi}{4} and x = \frac{1}{2}, then the particular solution of \frac{dy}{dx} = 2x\cos^2 y is
  33. 6(a)(i)7 marks· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 2Find the general solution of the differential equation (1 + x^2) \frac{\mathrm{d}y}{\mathrm{d}x} + 2xy = \sqrt[3]{x}.
  34. 6(a)(ii)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 2Hence, given that y = 2 when x = 0, calculate y(1).
  35. 6(b)(i)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 2Use the substitution u = y' to show that the differential equation y'' + 4y' = 2\cos 3x - 4\sin 3x can be reduced to u' + 4u = 2\cos 3x - 4\sin 3x.
  36. 6(b)(ii)13 marks· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 2Hence, or otherwise, find the general solution of the differential equation.
  37. Q341 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 1Given that y = 0 at x = 0, the general solution of the differential equation y'' + 6y' + 9y = 0 is
  38. Q401 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 1A suitable integrating factor for the solution of the differential equation \frac{dy}{dx} + \frac{2y}{x} = \frac{1}{x} \text{ is}
  39. Q411 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 1The general solution of the differential equation \frac{dy}{dx} = \frac{y}{x} is
  40. Q441 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 1The general solution of the differential equation \frac{d^2y}{dx^2} - 3\frac{dy}{dx} = 0 is
  41. 6(a)(i)9 marks· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 2Show that the general solution of the differential equation is y = (1/2) sec(x) - cos(x) + C sec(x).
  42. 6(a)(ii)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 2Hence, or otherwise, solve the initial value problem y' cos(x) = y sin(x) + sin(2x), y(0) = 0.
  43. 6(b)(i)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 2Determine the solution of the complementary equation y'' + 2y' + y = 0.
  44. 6(b)(ii)10 marks· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 2Given that the particular solution has the form y_p = (A x^3 + B x^2) e^(-x), or otherwise, determine the general solution of the differential equation.
  45. Q351 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 1Given that y = 0 at x = 0, the general solution of the differential equation y'' + 6y' + 9y = 0 is
  46. Q421 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 1If the auxiliary equation for a homogeneous second order differential equation with real, constant coefficients is given by \lambda^2 + 6\lambda + 50 = 0, then the general solution of the differential equation may be…
  47. Q431 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 1A suitable integrating factor for the solution of the differential equation \frac{dy}{dx} + \frac{2y}{x} = \frac{1}{x}\nis
  48. Q441 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 1Given that y = \frac{\pi}{4} and x = \frac{1}{2}, then the particular solution of \frac{dy}{dx} = 2x\cos^2 y is
  49. 6(a)9 marks· Pure Mathematics · Unit 2 Q6 6(a)Determine the general solution of the differential equation y' + y cot x = e²ˣ.
  50. 6(a)(i)5 marks· Pure Mathematics · Unit 2 Q6 6(a)(i)Show that the general solution of the differential equation is y = (c/x) - (2/x) cos x, where c is a constant.
  51. 6(a)(i)10 marks· Pure Mathematics · Unit 2 Q6 6(a)(i)Show that the general solution of the differential equation is y = 5/3 + 1/3 cos 2x + C cosec x, where C is a constant.
  52. 6(a)(ii)3 marks· Pure Mathematics · Unit 2 Q6 6(a)(ii)Hence, determine the particular solution of the differential equation that satisfies the condition y = 2 when x = π.
  53. 6(a)(ii)3 marks· Pure Mathematics · Unit 2 Q6 6(a)(ii)Hence, determine the particular solution given that y(π/2) = 0.
  54. 6(b)5 marks· Pure Mathematics · Unit 2 Q6 6(b)Determine the general solution of the differential equation dy/dx + (3/x)y = sin(2x)/x².
  55. 6(b)9 marks· Pure Mathematics · Unit 2 Q6 6(b)Solve the initial value problem x² dy/dx + 2xy = cos x, where y(π) = 0.
  56. 6(b)7 marks· Pure Mathematics · Unit 2 Q6 6(b)Show that the general solution of the differential equation dy/dx = (xy - y) / (x² - 4) is y = k(x - 2)^(1/4)(x + 2)^(3/4), where k is a constant.
  57. 6(b)(i)11 marks· Pure Mathematics · Unit 2 Q6 6(b)(i)Show that the general solution of the differential equation y'' - y' - 2y = 3e²ˣ is y = Ae²ˣ + Be⁻ˣ + xe²ˣ.
  58. 6(b)(i)8 marks· Pure Mathematics · Unit 2 Q6 6(b)(i)Solve the differential equation to obtain the general solution.
  59. 6(b)(i)7 marks· Pure Mathematics · Unit 2 Q6 6(b)(i)Determine the general solution of the differential equation y'' + 2y' + 5y = 0.
  60. 6(b)(ii)5 marks· Pure Mathematics · Unit 2 Q6 6(b)(ii)Hence, solve the differential equation given that at x = 0, y = 0 and y' = 7.
  61. 6(b)(ii)2 marks· Pure Mathematics · Unit 2 Q6 6(b)(ii)Hence, given that y = 1 when x = 0, determine the particular solution.
  62. 6(b)(ii)5 marks· Pure Mathematics · Unit 2 Q6 6(b)(ii)Hence, determine the solution of the boundary value problem y'' + 2y' + 5y = 0 with y(0) = 1, y'(π) = 2.
  63. 6(c)10 marks· Pure Mathematics · Unit 2 Q6 6(c)Solve the boundary-value problem y'' - y' - 2y = 0, given that when x = -1, y = 1 and when x = 1, y = 0.
  64. 6(c)12 marks· Pure Mathematics · Unit 2 Q6 6(c)Determine the general solution of the differential equation y'' - 7y' + 12y = sin x - cos x.
  65. 6(c)(i)6 marks· Pure Mathematics · Unit 2 Q6 6(c)(i)Show that the general solution of the differential equation is Ae^(3x/4) sin(√19x/4) + Be^(3x/4) cos(√19x/4), where A and B are constants.
  66. 6(c)(i)5 marks· Pure Mathematics · Unit 2 Q6 6(c)(i)Obtain the complementary function of the differential equation.
  67. 6(c)(ii)9 marks· Pure Mathematics · Unit 2 Q6 6(c)(ii)Hence, or otherwise, determine the particular solution of the differential equation given that at x = 0, y = 3 and y' = 0.
  68. 6(c)(ii)5 marks· Pure Mathematics · Unit 2 Q6 6(c)(ii)Determine a particular integral of the differential equation.
  69. 6(c)(iii)1 mark· Pure Mathematics · Unit 2 Q6 6(c)(iii)Hence, state the general solution of the differential equation.

Related topics in Counting, Matrices and Differential Equations