Differential equations — AS & A Level Further Mathematics (9231) questions by topic
First and second-order differential equations anchor this topic: solving a first-order linear equation with an integrating factor given a boundary condition, and finding particular solutions of second-order equations with constant coefficients driven by a sine or polynomial term.
Find, show and state are the top command words across 24 questions from 2023 to 2025 papers, worth from 7 to 14 marks, where selecting the correct trial particular-integral form for the forcing term is usually the deciding step. Working through each integrating-factor or particular-integral derivation with instant marking catches a mismatched trial form or a wrongly applied boundary condition before it invalidates the final exact-form answer.
May–June 2025, Paper 21
Find the particular solution of the differential equation 6 (d^2 x / dt^2) + 3 (dx/dt) + 6x = e^(-t), given that, when t = 0, x = dx/dt = 0.
Answer this question and get it marked →Find the solution of the differential equation dy/dx - ((x + 5)/(x^2 + 10x + 61)) y = 1, given that y = 0 when x = 3. Give your answer in an exact form.
Answer this question and get it marked →May–June 2025, Paper 22
Find the particular solution of the differential equation 6(d^2 x / dt^2) + 3(dx/dt) + 6x = e^(-t), given that, when t = 0, x = dx/dt = 0.
Answer this question and get it marked →Find the solution of the differential equation dy/dx - ((x + 5)/(x^2 + 10x + 61))y = 1, given that y = 0 when x = 3. Give your answer in an exact form.
Answer this question and get it marked →May–June 2025, Paper 23
Find the particular solution of the differential equation d^2x/dt^2 + dx/dt - 2x = 2t^2 + t - 1, given that, when t = 0, x = dx/dt = 0.
Answer this question and get it marked →Find the solution of the differential equation dy/dx - ((2x + 6)/(x^2 + 6x + 5))y = 4, given that y = 0 when x = 0. Give your answer in an exact form.
Answer this question and get it marked →May–June 2025, Paper 24
Find the particular solution of the differential equation d^2y/dx^2 + 4 dy/dx + 5y = 13 e^{3x} given that y = 1 and dy/dx = 0 when x = 0.
Answer this question and get it marked →Find the solution of the differential equation x dy/dx - y = 2x^2 tan^{-1} x for which y = (1/2)π when x = 1. Give your answer in the form y = f(x).
Answer this question and get it marked →May–June 2024, Paper 23
Find the general solution of the differential equation d^2x/dt^2 + 10 dx/dt + 25x = 338 sin t.
Answer this question and get it marked →October–November 2024, Paper 21
Find the particular solution of the differential equation 6 d^2x/dt^2 - 5 dx/dt + x = t^2 + t + 1, given that, when t = 0, x = 12 and dx/dt = -6.
Answer this question and get it marked →Show that an appropriate integrating factor for sqrt(x^2 + 16) dy/dx + y = x sqrt(x^2 + 16) is (1/4)x + (1/4)sqrt(x^2 + 16).
Answer this question and get it marked →October–November 2024, Paper 22
Find the particular solution of the differential equation 3 d^2y/dx^2 + 2 dy/dx + y = x^2, given that, when x = 0, y = dy/dx = 0.
Answer this question and get it marked →October–November 2024, Paper 23
Find the particular solution of the differential equation 6 d^2x/dt^2 - 5 dx/dt + x = t^2 + t + 1, given that, when t = 0, x = 12 and dx/dt = -6.
Answer this question and get it marked →Show that an appropriate integrating factor for sqrt(x^2 + 16) dy/dx + y = x sqrt(x^2 + 16) is (1/4)x + (1/4) sqrt(x^2 + 16).
Answer this question and get it marked →October–November 2024, Paper 33
A particle P of mass 2 kg moving on a horizontal straight line has displacement x m from a fixed point O on the line and velocity v m s^-1 at time t s. The only horizontal force…
Answer this question and get it marked →May–June 2023, Paper 21
Use the substitution z = x + y to find the solution of the differential equation dy/dx = (1 + 3x + 3y) / (3x + 3y - 1) for which y = 0 when x = 1. Give your answer in the form a…
Answer this question and get it marked →Find the particular solution of the differential equation d^2x/dt^2 - 12 dx/dt + 36x = 37 sin(t), given that, when t = 0, x = dx/dt = 0.
Answer this question and get it marked →May–June 2023, Paper 22
Use the substitution z = x + y to find the solution of the differential equation dy/dx = (1 + 3x + 3y) / (3x + 3y - 1) for which y = 0 when x = 1. Give your answer in the form…
Answer this question and get it marked →Find the particular solution of the differential equation d^2x/dt^2 - 12(dx/dt) + 36x = 37sin(t), given that, when t = 0, x = dx/dt = 0.
Answer this question and get it marked →May–June 2023, Paper 23
The variables x and y are related by the differential equation 6 d^2x/dt^2 + 5 dx/dt + x = t^2 + 10t + 13. Find the general solution for x in terms of t.
Answer this question and get it marked →October–November 2023, Paper 21
Find the particular solution of the differential equation d^2y/dx^2 + 2 dy/dx + 3y = 27x^2, given that, when x = 0, y = 2 and dy/dx = -8.
Answer this question and get it marked →October–November 2023, Paper 22
Find the solution of the differential equation dy/dx + 3y = sin x for which y = 1 when x = 0. Give your answer in the form y = f(x).
Answer this question and get it marked →It is given that v = y^4 and y^3 (d^2y/dx^2) + 3y^2 (dy/dx)^2 + y^3 (dy/dx) + y^4 = e^(-2x). Show that d^2v/dx^2 + dv/dx + 4v = 4 e^(-2x).
Answer this question and get it marked →October–November 2023, Paper 23
Find the particular solution of the differential equation d^2y/dx^2 + 2 dy/dx + 3y = 27 x^2, given that, when x = 0, y = 2 and dy/dx = -8.
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