In the past week, I studied Differential Equations through an MIT Online Open Course, which can be found here:
In this blog, I will introduce Differential Equations as well as some of the methods of solving or visualizing them. I will start from the place where most students left out since Calculus II to make it more comprehensible.
Differential Equations, also known as DE, means “an equation involving derivatives of a function of functions” (dictionary.com). Differential Equations have a broad application in subjects like Physics, Engineering, and Biology, which will be discussed in depth in my future blogs. The first several blogs, however, will be focusing on the math behind differential equations including how to solve and visualize the formulas.
One simple example of DE is , in which x is the independent variable and y is the dependent variable. Notice that taking integral is not a way to find a general solution of y; instead, we must employ a method called “Separation of Variables”.
This main concept of Separation of Variables is to separate the two variables, x and y, to two sides of the equation. For instance, would be rewritten as . Then, we can take integral on both sides and solve this differential equation.
Separation of Variables is a straightforward and neat method but is not applicable to many forms of differential equations. , for instance, cannot be solved using this method since x and y cannot be completely separated to two sides. This kind of differential equations is known as first-degree differential equations, which can be written as . As Separation of Variables cannot be used to find solutions for most first-degree differential equations, two other methods are introduced below.
The Geometric Method is one of the ways to solve or find properties of differential equations. The basic idea is to find sets of (x, y) to represent integral curves and plot them in a direction field. To understand the interrelation between the traditional analytic and geometric method, the following chart might be found helpful.
In the graph drawn by Desmos.com, the plot field of looks like:
The general solution of , which currently is unsolvable by hand, can be found on Scholastic.org in the form of , where c represents any constant. If we apply different values of c into the general solution, we should find the graph of that solution agree with the plot field.
Isocline is an important concept in the geometric method, especially when we need to draw the plot field by hand. An isocline is a line on which all the points share the same integral curve.
The above graph refers to the DE . In this graph from ck12.org, the red lines are the isoclines, and black line segments are the integral curves. On any of the isoclines, we can find the integral curves share the same direction and length. On k=-2, for instance, all the integral curves have a slope of -2.
To find an isocline, we can simply assign random values to , then solve the new equation. Take for instance, we would get from the equation, which matches with the bottom red straight line that passes through (2, 0) and (0, -2). In general: isocline is an intermediate step to help us construct a plot field, and a plot field would show the overall properties and shapes of the DE.
Euler’s Method a numerical method to estimate and analyze differential equations. As it is similar to the Euler’s Method in standard Calculus, this introduction will focus on the visualization of Euler’s Method rather than the algorithm.
In general, Euler’s Method is a process of chopping the continuous differential equations into many segments with the length on the x-direction being h.
Euler’s Method, however, doesn’t produce the accurate result, as we can easily see from the departing solution curve and the tangent line above. To decide if the result of Euler’s Method is larger or smaller than the actual value, we need to find the concavity of the DE. For a convex (concave up) curve, Euler’s Method produces a smaller result than the actual value; for a concave (concave down) curve, Euler’s Method produces a larger result than the actual value. The equations of Euler’s Method from (xn, yn) to (xn+1, yn+1) can be found as:
As h approaches 0, the result becomes more accurate. To comprehend this, the following graph might be found helpful.
This is the plot field of f(x,y)=y*sin(x). From the graph, we can easily see the larger the h, the farther the estimation is from the actual curve, which is represented by the blue color on the bottom.
Also notice that for some differential equations, a large step-size may result in the divergence between the estimated and actual value. As shown in the following graph, the result of h=1.00 (orange line) diverges from the actual curve while the result of h=0.125 (light blue line) efficiently estimates the curve.
CK-12. (2014, September 24). ODE: Solutions from Slope Fields and Isoclines. Retrieved September 9, 2018, from CK-12 website: https://www.ck12.org/calculus/slope-fields-and-isoclines/lesson/ODE:-Solutions-from-Slope-Fields-and-Isoclines/
Eddie. (2016, July 10). What is a solution to the differential equation d y d x = x − y ? Retrieved September 9, 2018, from https://socratic.org/questions/what-is-a-solution-to-the-differential-equation-dy-dx-x-y-2
Hohn, H., & Miller, H. (2001). EULER’S METHOD. Retrieved September 9, 2018, from Applet courtesy of MIT MATHLETS website: https://ocw.mit.edu/ans7870/18/18.03SC/eulersMethod.html
Slope Field Generator. (n.d.). Retrieved September 9, 2018, from Desmos website: https://www.desmos.com/calculator/p7vd3cdmei
Arthur Mattuck, Haynes Miller, Jeremy Orloff, and John Lewis. 18.03SC Differential Equations. Fall 2011. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA.