Contents
- 1 Introduction to Differential Equations
- 1.1 1.0 Abstract
- 1.2 1.1 Concept
- 1.3 1.2 Ordinary Differential Equations (ODE) & Partial Differential Equations (PDE)
- 1.4 1.3 Usual Notations for Differential Equations
- 1.5 1.4 Terminologies for Differential Equations
- 1.6 1.5 The Solutions to Differential Equations
- 1.7 1.6 Exercises
- 1.8 1.7 References
Introduction to Differential Equations
1.0 Abstract
When I was first learning differential equations myself, I felt hard to understand the textbook and my lecture notes. After winning the battle, now I am trying to build up those concepts again and explain them in an easier and more concise way. It is not that academic, aiming for intuitive understanding.
1.1 Concept
Previously, we learnt to solve algebraic equations with numbers as solutions. For example, for linear equations $ ax+b=0 $ with respect to $ x $ and $ a≠0 $, the solution is going to be $ x=-\frac{b}{a} $. For quadratic equations $ ax^2+bx+c=0 $ with respect to $ x $ and $ a≠0 $, the solution is going to be $ x={\frac{-b±\sqrt{b^2-4ac}}{2a}} $.
In both examples, the solutions $ x=-\frac{b}{a} $ and $ x={\frac{-b±\sqrt{b^2-4ac}}{2a}} $ are particular numbers, and mostly we are discussing them within real numbers. When a set of numbers match another set of numbers by a particular definition, a function forms. Therefore, if we apply the function to the solution, (i.e. the equation has a function as a solution) a differential equation forms. Consider what we have learned in Calculus I. When we are taking derivatives to a function, the result is still going to be a function (either constant functions or not).
Here comes the formal concept. A differential equation is an equation involving derivatives, with function(s) as its solution(s).
1.2 Ordinary Differential Equations (ODE) & Partial Differential Equations (PDE)
From 1.1, we have built up the concept of involving derivatives to an equation. As we know, derivatives also involve ordinary and partial derivatives. This separates the differential equations to ODE and PDE. Here is a brief comparison.
| ODE | PDE | |
|---|---|---|
| Number of independent variables | 1 | More than 1 |
| How does is look like | $ \frac{dy}{dx}=y^2+y $, $ k $ is a parametre | $ \frac{dh}{dt}=k\frac{d^2h}{dx^2} $, $ k $ is a parametre |
| Dependent variable: $ y $ | Dependent variable: $ y $ | |
| Independent variable: $ y $ | Independent variables: $ x $, $ y $ |
In this tutorial, we are mainly focusing on ordinary differential equations (ODE).
1.3 Usual Notations for Differential Equations
| “Dot” usually refers to taking derivatives w.r.t. $ t $ | “Prime” usually refers to taking derivatives w.r.t. $ x $ | |
|---|---|---|
| Examples ..... | $ ü=\frac{d^2u}{dt^2} $ | $ y'=\frac{dy}{dx} $ |
Remark: “w.r.t.” refer to “with respect to”
1.4 Terminologies for Differential Equations
· Order: The order of a differential equation is the highest derivative involved.
E.g. $ \frac{dx}{dt}=tx $: 1st order, $ \frac{d^2x}{dt^2}=kx $: 2nd order, ..., $ \frac{d^nx}{dt^n}=8x $: nth order
· Linearity: An nth-ordered ODE for $ y(t) $ is linear, if it can be written as $ a_n(t)\frac{d^ny}{dx^n}+a_{n-1}(t)\frac{d^{n-1}y}{dx^{n-1}}+...+a_1(t)\frac{dy}{dt}+a_0(t)y=f(t) $, where $ f(t) $ is a known function. Otherwise, it is non-linear.
More intuitively, a differential equation is linear if there is not a multiplication term of the variables (either dependent or independent) and their differential terms (e.g. $ x\frac{dy}{dx} $, $ y\frac{dy}{dx} $ ).
· Homogeneity: In $ a_n(t)\frac{d^ny}{dx^n}+a_{n-1}(t)\frac{d^{n-1}y}{dx^{n-1}}+...+a_1(t)\frac{dy}{dt}+a_0(t)y=f(t) $, if $ f(t)=0 $, the equation is homogeneous. If $ f(t)≠0 $, the equation is non-homogeneous.
Remark: If a differential equation is homogeneous, then we can derive more solutions to the equation by multiplying constants to one solution, or adding up some of its solutions.
1.5 The Solutions to Differential Equations
As we know from 1.1, the solutions to differential equations are functions. In terms of the format of expression, there are two kinds of functions, explicit functions and implicit functions. Explicit functions are what we learnt at the very beginning, in which they express the dependent variable by the independent variable, e.g. $ x=sint+cost $. On the contrary, implicit functions are those with a dependent variable that is super hard to separate out from an equation, e.g. $ lnx+x^5+3t-5=0 $.
Hence, there are also two kinds of solutions to differential equations, explicit (e.g. $ x=sint+cost $) and implicit solutions (e.g. $ lnx+x^5+3t-5=0 $).
1.6 Exercises
1. Which of the followings is (are) differential equation(s)?
A. $ 4x^5+2\sqrt{x}=0 $
B. $ ü=t^2+t $
C. $ lnx+x^5+3t-5=0 $
D. $ \frac{d^2x}{dx^2}+\frac{dx}{dt}+3t=0 $
2. Which of the followings is (are) ordinary differential equation(s)?
A. $ y'=x^2+x $
B. $ \frac{dh}{dt}+3t=\frac{d^2h}{dx^2}-2x $
C. $ \frac{dx}{dt}=x^2+x $
D. $ 4x^5+2\sqrt{x}=0 $
3. Indicate the independent variables and dependent variables of the following differential equations, and then find their orders, linearity and homogeneity.
· $ \frac{dy}{dx}=\frac{y(x+siny)}{xy*cosx-5} $
· $ x''-5x'+4x=e^{2t} $
· $ xy\frac{dy}{dx}=2x^2+3y^2 $
4. Show that $ x^2+xcosy+lny=7 $ is an implicit solution to the differential equation $ \frac{dy}{dx}=\frac{y(2x+cosy)}{xy*siny-1} $
1.7 References
Institute of Natural and Mathematical Science, Massey University. (2017). 160.204 Differential Equations I: Course materials. Auckland, New Zealand.
Robinson, J. C. (2003). An introduction to ordinary differential equations. New York, NY., USA: Cambridge University Press.
Zill, D. G., Wright, W. S., & Cullen, M. R. (2013). Differential equations: with boundary-value problems. (8th ed.). Boston, MA., USA: Brooks Cole/Cengage Learning.
