Differential equations — AS & A Level Mathematics (9709) questions by topic
Separating variables to solve an equation linking x and y with a given initial condition, working through a trigonometric differential equation for x in terms of theta, and modelling a cuboid water tank filling and leaking simultaneously are typical of the 14 differential equations questions from 2023 to 2025, worth between 7 and 13 marks.
Solve, show and find lead the command words, with state used for writing the differential equation itself before it is solved. Because separating variables incorrectly or mishandling a constant of integration derails the entire solution, submit your working for instant marking, so a dropped constant or a wrongly separated variable is caught the moment it appears rather than after the final expression fails to satisfy the given condition.
May–June 2025, Paper 32
The variables x and theta satisfy the differential equation sin 2theta (dx/dtheta) = (4x + 3) cos 2theta, and x = 0 when theta = (1/12)pi. Solve the differential equation and…
Answer this question and get it marked →May–June 2025, Paper 33
The variables x and y satisfy the differential equation sin 4y (dy/dx) = x sin 2y sin 3x. It is given that y = (1/12)π when x = (1/2)π. Solve the differential equation, obtaining…
Answer this question and get it marked →May–June 2025, Paper 35
The variables x and y satisfy the differential equation (x^2 + 3)(dy/dx) = e^(3y)(x − 2). It is given that y = 0 when x = 0. Solve the differential equation, and find the value of…
Answer this question and get it marked →May–June 2024, Paper 31
In a field there are 300 plants of a certain species, all of which can be infected by a particular disease. At time t after the first plant is infected there are x infected…
Answer this question and get it marked →May–June 2024, Paper 33
A container in the shape of a cuboid has a square base of side x and a height of (10 − x). It is given that x varies with time, t, where t 0. The container decreases in volume at…
Answer this question and get it marked →October–November 2024, Paper 31
A large cylindrical tank is used to store water. The base of the tank is a circle of radius 4 metres. At time t minutes, the depth of the water in the tank is h metres. There is a…
Answer this question and get it marked →October–November 2024, Paper 32
A balloon in the shape of a sphere has volume V and radius r. Air is pumped into the balloon at a constant rate of 40π starting when time t = 0 and r = 0. At the same time, air…
Answer this question and get it marked →October–November 2024, Paper 33
A water tank is in the shape of a cuboid with base area 40 000 cm². At time t minutes the depth of water in the tank is h cm. Water is pumped into the tank at a rate of 50 000 cm³…
Answer this question and get it marked →May–June 2023, Paper 31
The variables x and y satisfy the differential equation cos 2x (dy/dx) = 4 tan 2x / sin² 3y, where 0 ⩽ x < ¼π. It is given that y = 0 when x = ⅙π. Solve the differential equation…
Answer this question and get it marked →May–June 2023, Paper 32
The variables x and y satisfy the differential equation dy/dx = (4 + 9y²) / e^(2x+1). It is given that y = 0 when x = 1. Solve the differential equation, obtaining an expression…
Answer this question and get it marked →May–June 2023, Paper 33
The variables x and y satisfy the differential equation dy/dx = (y² + 4) / (x(y + 4)) for x 0. It is given that x = 4 when y = 2√3. Solve the differential equation to obtain the…
Answer this question and get it marked →October–November 2023, Paper 31
The variables x and θ satisfy the differential equation (x / tan θ) · (dx/dθ) = x² + 3. It is given that x = 1 when θ = 0. Solve the differential equation, obtaining an expression…
Answer this question and get it marked →October–November 2023, Paper 32
The variables x and y satisfy the differential equation x² (dy/dx) + y² + y = 0. It is given that x = 1 when y = 1. Solve the differential equation to obtain an expression for y…
Answer this question and get it marked →October–November 2023, Paper 33
The variables x and y satisfy the differential equation e^(4x) · dy/dx = cos²3y. It is given that y = 0 when x = 2. Solve the differential equation, obtaining an expression for y…
Answer this question and get it marked →Other AS & A Level Mathematics topics
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