1. MAT 3237 Differential Equations Section 18.4 Series Solutions Part I http://myhome.spu.edu/lauw

2. HW No WebAssign Do18.4 #1 Do not turn in

3. Why Power Series Solutions? Many DE cannot be solved explicitly in terms of finite combinations of elementary functions.

4. Example Quantum Mechanics

5. Example Quantum Mechanics

6. Extract Information (e.g. from Approximation) Quantum Mechanics

7. Review Power Series Differentiate Power Series

8. Recall: Index Shifting Rules

9. decrease the index by 1 increase the i in the summation by 1

10. Recall: Index Shifting Rules increase the index by 1 decrease the i in the summation by 1

11. Example

12. Definition A Power Series is of the form

14. Theorem (Identity Property)

15. Theorem

18. Example 1 Solve the following DE by power series

19. Example 1 Solve the following DE by power series

20. Expectation

21. Summary To combine the summations, we need same power of x and same index ranges Use different representations of the derivative index shifting Find the relationship between the coefficients and look for patterns do not “collapse” numbers The lowest coefficient(s) remains (fix by I.C.)