Euler's method is a numerical approximation algorithm that helps in providing solutions to a differential equation. When used by a computer, the algorithm provides an accurate represntation of the solution curve to most differential equations..
1. Graph generation: given an initial point, a differential equation, and a step size, the solution curve of the differential matching the initial point is generated through Euler's method. More on how Euler's method works can be found in the corresponding video on this website
2. Estimating y at x: since a collection of Euler points, sorted according to their x-values, is generated and then connected to obtain the shape of the graph, it is possible to estimate the value of y at any value x = x0 in between two Euler points by finding two neighbouring Euler points P(x1,y1) and Q(x2,y2) such that x0 ∈ (x1,x2). Afterwards, based on simple linear equations, it is known that the in order for the point at x = x0 to be on the solution curve, it must be on the line segment formed by the two Euler points P and Q. Therefore, the corresponding y value at x = x0 is
y0 = y1 + (x0 － x1) · y0 － y1⁄x2 － x1
3. Finding solutions for x at y: similar to the "Estimating y at x" feature, in order to find the x-values at y = y0, the collection of Euler points is scanned through and every time two neighbouring points P(x1,y1) and Q(x2,y2) and are found such that y0 ∈ (y1,y2), then based on linear equations, it is known that as the solution point (x0,y0) is on the graph, then point (x0,y0) must also be on the line segment formed by points P and Q. Therefore, the corresponding x value at y = y0 is
x0 = x1 + (x2 － x1) · y0 － y1⁄y2 － y1
Differentials are an inseparable part of physics (as observed in kinematics and electricity). Sometimes, the differentials that exist naturally in physics can be unsolvable given our current understanding of differentials. The following example illustrates this.
Problem and solutions slideshow
Mixing problems in general have many applications, such as this plant nutrition problem that is found in the PDF below. Quite often, the differentials we get when solving day-to-day problems are not as easy to solve, and again, Euler's method is a tool which can be used to help obtain the solutions.
Problem and detailed solutions
The following problem connects concepts learnt in calculus to practical applications in engineering and statistics. This question is a real-life example of problems that engineers face in their day-to-day work.
Source code avaliable on GitHub