Euler’s method: notes and practice questions

Consider the following differential equation:

$\left( 2x^{3} + 5 \right)\frac{dy}{dx} = 6x^{2}y^{2}$

$a$ Take $y( - 1) = - 1$, and use Euler's method with a step size of 0.4 to estimate $y$ at $x = 1$.

$b$ By solving the differential equation analytically, calculate the absolute error in your approximation from part (a) at $x = 1.$

(a) In a controlled biological experiment, the rate of change of the population $P$ of a certain microorganism with respect to time $t$ is modeled by the differential equation $t^2 \frac{dP}{dt} = P^2 - 2tP + 2t^2$, where $t > 0$ is in hours and $P$ is in thousands of organisms. It is known that at $t = 1$ hour, the population is $P = 4$ thousand.

Use Euler's method, with a step length of 0.1, to find an approximate value of $P$ when $t = 1.4$.

(b) Use the substitution $P = vt$ to show that $t\frac{dv}{dt} = v^2 - 3v + 2$.

(c.i) By solving the differential equation from part (b), and given that $P > 2t$, show that $P = \frac{6t - 2t^2}{3 - 2t}$.

(c.ii) Find the actual value of $P$ when $t = 1.4$.

(c.iii) Using the graph of $P = \frac{6t - 2t^2}{3 - 2t}$, suggest a reason why the approximation given by Euler's method in part (a) is not a good estimate to the actual value of $P$ at $t = 1.4$.

A metal object is cooling in a room. Its temperature, $T$ (in degrees Celsius), at time $t$ (in minutes) can be modelled by the differential equation $\frac{dT}{dt} = -0.05(T-20)$. Initially, at $t=0$, the temperature of the object is $80^{\circ}\text{C}$.

(a) Use Euler's method with a step size of 0.1 to find an approximation for the temperature of the object when $t=0.4$ minutes. Give your answer correct to four significant figures.

(b) By solving the differential equation, show that $T = 20 + 60e^{-0.05t}$.

(c) Find the absolute value of the error in your approximation in part (a).

The rate of change of the concentration, $C$, of a reactant in a chemical process at time $t$ is modelled by the differential equation $\frac{dC}{dt} = 2t - \frac{C}{t}$, for $t > 0$. Given that the initial concentration is $C(1) = 2$, use Euler's method with a step size of $h = 0.2$ to find an approximate value of $C$ when $t = 1.6$.

Solve the differential equation $\frac{dC}{dt} = 2t - \frac{C}{t}$ using an appropriate analytical method, and find the exact value of $C$ when $t = 1.6$.

Sketch the approximate values you found in part (a) and the graph of the function you found in part (b) on the same coordinate axes. Use your sketch to explain why your answer in part (a) is greater than or less than the value in part (b).