Differential Equations: Lecture 1

The term differential equation was coined by Leibniz in 1676 for a relationship between the two differentials dx and dy for the two variables x and y.

A differential equation is an equation which involves an unknown function (e.g. y) and some of its derivatives. Solving a differential equation (DE) is to find a relation between the variables, say y and x, that satisfies the DE.

In today's lecture, we have learnt the followings:

1. Terminology: what is the order and degree of a differential equation?

2. How to solve the following two types of DEs?

• \[\frac{\mathrm{d} y}{\mathrm{d} x}=f(x)\]

To get the solution, we use \[y=\int f(x) dx\]

• \[\frac{\mathrm{d} y}{\mathrm{d} x}=f(y)\]

To get the solution, we use \[\int \frac{1}{f(y)} dy=x\]

3. General solutions and particular solutions

A differential equation has infinite no. of solutions. For example, \[y=x^{2}+1,y=x^{2}+2 \] are both the solutions for the DE \[\frac{\mathrm{d} y}{\mathrm{d} x}=2x\]. Indeed, \[y=x^{2}+C\] will be a solution for the DE with any constant C.

In this case, we call \[y=x^{2}+C\] a general solution.

If a condition on x and y is given, say y=1 when x=0, then we can find out a specific value of C and that is a particular solution.

After today's lecture, you can attempt Q1-Q7 in tutorial 11a.