Differential Equations & Rates — practice questions

Solve $\dfrac{dy}{dx} = 4x - 6$ given that the curve passes through $(3, 1)$.

Solve $\dfrac{d^2y}{dx^2} = e^{2x}$ given that $\dfrac{dy}{dx} = 1$ and $y = 0$ when $x = 0$.

A curve satisfies $\dfrac{d^2y}{dx^2} = \dfrac{6}{x^3}$ for $x > 0$. It has a stationary point at $(1, 4)$. Find the equation of the curve, and determine whether the stationary point is a maximum or a minimum.

Solve $\dfrac{dy}{dx} = 3y$ given that $y = 8$ when $x = 0$.

Solve $\dfrac{dy}{dx} = \dfrac{2x}{y^2}$ given that $y = 2$ when $x = 1$, giving $y$ explicitly in terms of $x$.

A population $P$ grows according to $\dfrac{dP}{dt} = kP$. It is 4000 initially and 6500 after 3 years. Find $k$ exactly, find the population after 10 years, and determine when the population will double.

A radioactive sample decays according to $A = A_0e^{-0.05t}$, where $t$ is in days. If the initial mass is 200 g, find the mass after 12 days.

A bacterial culture grows so that its rate of increase is proportional to the number present. There are 500 bacteria initially and 1800 after 4 hours. Find the number after 10 hours.

A body is found at 11:00 pm with a temperature of 30 °C. One hour later it is 28 °C. The room has been at a constant 20 °C. Assuming Newton's law of cooling and a normal body temperature of 37 °C, estimate the time of death and discuss the reliability of the estimate.

A particle has acceleration $a = 6t - 2$ m s⁻². Its initial velocity is 4 m s⁻¹. Find its velocity when $t = 3$ s.

A particle moves with velocity $v = 4e^{-2t}$ m s⁻¹. Its displacement is 0 when $t = 0$. Find the displacement function, and find the limiting displacement as $t \to \infty$.

Water flows into a tank at a rate of $\dfrac{dV}{dt} = \dfrac{20}{t + 1}$ litres per minute, and simultaneously leaks out at a constant 2 litres per minute. The tank is empty at $t = 0$. Find the maximum volume of water in the tank and the time at which it occurs, and determine when the tank is empty again.