Learning Outcomes:

1. SECOND ORDER DIFFERENTIAL EQUATIONS Learning Outcomes: • Know the general form of a second order differential equations • Know the difference between homogeneous and non-homogeneous • homogeneous • non-homogeneous • Be able to solve homogeneous second-order differential equations

2. Solving homogeneous second order differential equations - STRATEGY Step 1: Write down the auxiliary equation Step 2: Solve auxiliary equation. (find m1 and m2) Step 3: Write down the general solution depending on the nature of the roots of the auxiliary equation • real and distinctm1 and m2 • real and coincident (double root) m • complex p ± iq

3. Example: Find the general solution of P118 Step 1: Auxiliary equation Step 2: Solve auxiliary equation. real distinct roots Step 3: Write down the general solution

4. Example: Find the particular solution of given that when P118 Step 1: Auxiliary equation Step 2: Solve auxiliary equation. real distinct roots Step 3: General solution Page 119 Ex 3 Step 4: Use initial conditions to find A and B Particular Solution

5. Example: Find the general solution of Step 1: Auxiliary equation Step 2: Solve auxiliary equation. real coincident roots Step 3: Write down the general solution

6. Example: Find the particular solution of given that when and when P120 Step 1: Auxiliary equation real coincident roots Step 2: Solve auxiliary equation. Step 3: General solution Page 120 Ex 4 Step 4: Use initial conditions to find A and B Particular Solution 1

7. Example: Find the general solution of P121 Step 1: Auxiliary equation Step 2: Solve auxiliary equation. complex roots Step 3: Write down the general solution

8. Example: Find the particular solution of given that when P121 complex roots Step 1: Auxiliary equation Step 2: Solve auxiliary equation. Step 3: General solution Step 4: Use initial conditions to find A and B 0 0 0 2 Page 122 Ex 5 Particular Solution

9. If and are solutions, so are , , & • If and are two independent solutions, then Finding and is the general solution of Solving homogeneous second order differential equations – Explanation Important points • General solution of 2nd order equation contains two arbitrary constants.

10. is the general solution of • Finding and is a solution of • Check if it is a solution of • Solving the auxiliary equation gives us two values if • which gives us two solutions and • real and distinctm1 and m2 Solving homogeneous second order differential equations – Explanation Real distinct roots auxiliary equation general solution

11. is the general solution of • Finding and • When we only get one value of m which gives us one solution We can show that a second solution is • real and coincidentm Solving homogeneous second order differential equations – Explanation Real coincident roots auxiliary equation • We need a second proof page 119 general solution

12. REMINDER – Power Series

13. is the general solution of • Finding and • When we get complex conjugates as values of m • complex conjugatesp + iq Solving homogeneous second order differential equations – Explanation Complex roots auxiliary equation general solution