Differential Equation: Lecture 2

In lecture 2, we learnt two main skills.

1. How to solve a second order DE in the form of \[\frac{\mathrm{d^2}y }{\mathrm{d} x^2}=f(x)\]

2. How to use substitution to solve a first order DE that cannot be solved by direct integration.


1. How to solve a second order DE

Step 1: integrate both sides once to get a first order DE (the first order DE obtained will have one arbitrary constant.)

Step 2: integrate again to get the general solution. (the general solution will have 2 arbitrary constant.)

2. How to use substitution to solve a first order DE that cannot be solved by direct integration.

Step 1: differentiate the given formula for substitution. (e.g. z=1/y)

Step 2: replace the dy/dx in the original DE by dz/dx

Step 3: observe whether there are still y in the DE, replace the y using substition or other methods.

The final DE should be in term of x and z and should be solvable by direct integration.

Step 4: solve the DE to obtain a relation between x and z.

Step 5: replace z by y using the substition given in the question.

After lecture 2, you can do the whole tutorial 11a.