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Further Pure 3: Differential Equations Past Examination Questions 1 (i) Show that by using the substitution y 1 , the differential equation z dy 2 y xy 2 dx may be written in the form dz 2z x . dx (ii) [3] Find the general solution of dz 2z x , dx and hence find the general solution of dy 2 y xy 2 . dx [5] OCR P4 May 2005 2. (i) (ii) 3. (i) (ii) 4 (i) The movement of the needle on an instrument used for measuring the strength of an electrical current can be modelled by the differential equation d 2 d 4 2 5 0 , dt dt where θ represents the angle turned through from a standard position and t represents time. Find the general solution of the differential equation. [3] Initially the needle is at rest in a position corresponding to . Express in 4 [4] terms of t. d 0 when t = 0]. [Hint: If the needle is initially at rest, this means that dt OCR P5 June 2005 Find the general solution of the differential equation d2y dy 6 10 y 10. [5] 2 dx dx Find the particular solution representing a curve which has tangent y = x at the point (0, 0). [6] dy 1 when x = 0 (since y = x has [Hint: This means that y = 0 when x = 0 AND dx gradient 1). OCR P5 January 2005 Find the values of the constants λ and μ such that y x sin 3x x cos 3x is a particular integral of the differential equation d2y 9 y sin 3x . dx 2 [5] (ii) Find the particular solution of the differential equation d2y 9 y sin 3x dx 2 which has a stationary value at the point 16 ,0 . [6] OCR P5 June 2004 5 (i) (ii) 6 Show that an integrating factor of the differential equation dy xy x dx 1 x 2 [3] is (1 x 2 ) . Hence solve the differential equation, given that y = 1 when x = 0, giving your answer in the form y = f(x). [5] OCR P4 May 2004 Find the general solution of the differential equation dy y e3x , dx giving your answer in the form y = f(x). [4] OCR P4 January 2005 7. It is given that y satisfies the differential equation d2y dy 5 4 y 8 x 10 10cos 2 x . 2 dx dx [3] a) Show that y 2 x sin 2 x is a particular integral of the given differential equation. b) Find the general solution of the differential equation. [4] [4] dy c) Hence express y in terms of x, given that y = 2 and = 0 when x = 0. dx AQA FP3 June 2006 8. a) Show that sinx is an integrating factor for the differential equation dy (cot x) y 2 cos x . [3] dx b) Solve this differential equation, given that y = 2 when x . [6] 2 AQA FP3 June 2006 9. [2] a) Find the roots of the equation m2 2m 2 0 in the form a + ib. b) (i) Find the general solution of the differential equation d2y dy 2 2 y 4x . [6] 2 dx dx [4] dy (ii) Hence express y in terms of x, given that y = 1 and = 2 when x = 0. dx AQA FP3 January 2006 10. a) Show that y x3 x is a particular integral of the differential equation dy 2 xy 2 5x2 1 . dx x 1 [3] b) By differentiating x 2 1 y c implicitly, where y is a function of x and c is a c is a solution of the differential equation x 1 dy 2 xy [3] 2 0. dx x 1 c) Hence find the general solution of dy 2 xy 5x2 1 . [2] dx x 2 1 AQA FP3 January 2006 constant, show that y 11 (i) (ii) (iii) 2 1 is an integrating factor for the first-order differential equation x dy 1 y x ln x . [3] dx x Solve this differential equation, given that y = 1 when x = 1. [6] Calculate the value of y when x = 1.2, giving your answer to three decimal places. [1] AQA FP3 January 2006 Show that 12. a) Find the integrating factor for the differential equation [2] dy y ex . dx b) Find the general solution of this differential equation, giving your answer in the form y = f(x). [4] AQA P5 June 2005 13. Find the general solution of the differential equation d 2 y dy 2 y 3cos x 4sin x . dx 2 dx [11] AQA P5 June 2005 14. a) Obtain the roots of the equation m 2 4m 8 0 , giving your answers in the form a ib . b) Solve the differential equation d2y dy 4 8 y 8e2 x 2 dx dx dy 2 when x = 0. given that y = 2 and dx [2] [12] AQA P5 January 2005 15 1 dy dz , where z is a function of x, express in terms of z and . z dx dx b) It is given that y satisfies the differential equation dy x2 yx y 2 , x 0 . dx 1 Show that the substitution y transforms this equation into the differential z equation a) Given that y [1] c) dz z 1 2 . dx x x (i) Obtain the general solution of the differential equation dz z 1 2 . dx x x [2] [5] [3] (ii) Hence obtain y in terms of x, given that y = 2 when x = ½. AQA P5 January 2005 16.(a) (a) Find the general solution of the differential equation dy d2 y +2 + 2y = 2et. 2 dt dt (b) Find the particular solution that satisfies y = 1 and 17.(a) [6] dy = 1 at t = 0. [6] dt EDEXCEL June 2004 P4 (a) Find the general solution of the differential equation dy + 2y = x. dx [5] Given that y = 1 at x = 0, (b) find the exact values of the coordinates of the minimum point of the particular [4] solution curve, [2] (c) draw a sketch of this particular solution curve. EDEXCEL June 2004 P4 18. dy y 1 dx 3 1 , x > 0. x x2 (a) Verify that x3ex is an integrating factor for the differential equation. [3] (b) Find the general solution of the differential equation. [4] (c) Given that y = 1 at x =1, find y at x = 2. [3] EDEXCEL January 2004 P4