The crossing black lines are for three values of the first constant the three colours are for three values of the second constant. Here there are two solutions running through each point in the x,y plane. This equation has two integration constants. You can obtain the solution by integrating twice:d x2 y( ) x) k + C1 ( 2 dx x3 y (x) k + C1 x + C2 ( ) 6 The OrderThe following equation contains a second derivative. In this simple example a single solution runs through each point of the x-y plane. The plot here shows some particular solutions. If you give C some value, you have a particular solution. The set is known as the general solution. This can have any value - the equation has an infinite number of solutions. It does contain an integration constant C. This solution is a function that does not contain any derivative. We can integrate both sides to get:y x2 yk + C 2 The SolutionThe solution to our equation is most easily obtained using differential notation:dy k x dx We will use all these notations - choosing the simplest for a given problem. A derivative is often written using prime notation y'(x) and in this subject also using the symbol p (parameter):d dy ( y( ) x) y (x) p ( ) dx dx The derivative can be regarded as the quotient of two differentials dx and dy. The left side of the equation only contains a first derivative: this gives a first order equation. The other variable x is the independent one. We will regard y as a function of x - it is the dependent variable. It contains two variables x and y and a constant k. Some TermsHere is a simple differential equation: Hans Wesselingh, Groningen, April 2012Ĭontents 1 Introduction (Terminology) 2 First Order, First Degree Equations 3 First Order, Higher Degree Equations 4 Second Order, Linear Equations 5 Laplace Transforms 6 Numerical Methods References Hans Wesselingh and Hans de Waard, Calculate and Communicate with Mathcad Prime, VSSD Publishers, Delft 2012 (Erwin Kreyszig, Advanced Engineering Mathematics, John Wiley & Sons (many editions)ġ Introduction (Terminology)In this first lesson, we look at names and terms in differential equations. They only contain the basics - no engineering examples and no exercises. These are my notes on ordinary differential equations, worked out in Mathcad Prime 2.0.
Wesselingh, professor emeritus Department of Chemical Engineering University of Groningen
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