L3 Integration
Question Bank
Solve the differential equation
$ \frac{dy}{dx} = 6x^2y $
given that $y = 3$ when $x = 1$.
Hence find the value of $y$ when $x = 0.5,$ correct to 3 significant figures.
You must use calculus and show the results of any integration needed to solve the problem.
Solve the differential equation
$ \frac{dy}{dx} = y\cos 2x $
given that $y = 2$ when $x = \dfrac{\pi}{4}$.
Hence find the exact value of $y$ when $x = \dfrac{\pi}{12}$.
You must use calculus and show the results of any integration needed to solve the problem.
Solve the differential equation
$ \frac{dy}{dx} = \frac{2xe^{x^2}}{y} $
given that $y = 2$ when $x = 0$.
Hence find the value of $y$ when $x = 1,$ correct to 3 significant figures.
You must use calculus and show the results of any integration needed to solve the problem.
Solve the differential equation
$ \frac{dy}{dx} = \frac{\sec^2 x}{y^2} $
given that $y = 2$ when $x = \dfrac{\pi}{4}$.
Hence find the value of $y$ when $x = \dfrac{\pi}{3},$ correct to 3 significant figures.
You must use calculus and show the results of any integration needed to solve the problem.
Solve the differential equation
$ \frac{dy}{dx} = (2x + 1)e^{-y} $
given that $y = 0$ when $x = 1$.
Hence find the exact value of $y$ when $x = 3$.
You must use calculus and show the results of any integration needed to solve the problem.
Solve the differential equation
$ \frac{dy}{dx} = \frac{\sqrt{y}}{x + 1}, \quad x > -1 $
given that $y = 4$ when $x = 0$.
Find the exact value of $x$ when $y = 9$.
You must use calculus and show the results of any integration needed to solve the problem.
Solve the differential equation
$ \frac{dy}{dx} = \frac{3x^2}{y(x^3 + 1)}, \quad x > -1 $
given that $y = -2$ when $x = 0$.
Hence find the exact value of $y$ when $x = 2$.
You must use calculus and show the results of any integration needed to solve the problem.
Solve the differential equation
$ \frac{dy}{dx} = y^2\cos x $
given that $y = \dfrac{1}{2}$ when $x = 0$.
Hence find the greatest value of $y$.
You must use calculus and show the results of any integration needed to solve the problem.
Solve the differential equation
$ \frac{dy}{dx} = 2x(y - 1) $
given that $y = 3$ when $x = 0$.
Hence find the values of $x$ for which $y = 7$. Give your answers in exact form.
You must use calculus and show the results of any integration needed to solve the problem.
Solve the differential equation
$ \frac{dy}{dx} = \frac{2\sin^2 x}{y} $
given that $y = 2$ when $x = 0$.
Hence find the exact value of $y$ when $x = \dfrac{\pi}{2}$.
You must use calculus and show the results of any integration needed to solve the problem.
A jet boat's engine is cut when the boat is travelling at $20$ m/s. As it coasts, its velocity, $v$ m/s, $t$ seconds after the engine is cut, satisfies
$ \frac{dv}{dt} = -0.01v^2 $
(a) Find how long it takes the boat to slow down to $5$ m/s.
(b) Find how far the boat travels in that time. Give your answer correct to 3 significant figures.
You must use calculus and show the results of any integration needed to solve the problem.
Solve the differential equation
$ (x^2 + 1)\frac{dy}{dx} - 2xy - 2x = 0 $
given that $y = 3$ when $x = 1$. Give $y$ in terms of $x$.
You must use calculus and show the results of any integration needed to solve the problem.
Solve the differential equation
$ \frac{dy}{dx} = xy + y - 2x - 2 $
given that $y = 5$ when $x = 0$.
Hence find the exact minimum value of $y$.
You must use calculus and show the results of any integration needed to solve the problem.
A curve passes through the point $(0, -\ln 3)$ and satisfies the differential equation
$ \frac{dy}{dx} = 2xe^{x^2 + y} $
Find the equation of the curve, giving $y$ in terms of $x$.
The curve has two vertical asymptotes. Find their equations.
You must use calculus and show the results of any integration needed to solve the problem.
The graph shows a closed, egg-shaped curve that passes through the origin, O.
At every point on the curve where $y \ne 0,$
$ \frac{dy}{dx} = \frac{2 - e^x}{y} $
Find the exact coordinates of the lowest point on the curve.
You must use calculus and show the results of any integration needed to solve the problem.
Solve the differential equation
$ x\ln x \, \frac{dy}{dx} = y^2, \quad x > 1 $
given that $y = 1$ when $x = e$.
Hence find the exact value of $x$ for which $y = 2$.
You must use calculus and show the results of any integration needed to solve the problem.