MAT 3237 Differential Equations Section 18.4 Series Solutions Part I

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

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

Example Quantum Mechanics

Example Quantum Mechanics

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

Review Power Series Differentiate Power Series

Recall: Index Shifting Rules

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

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

Example

Definition A Power Series is of the form

Theorem (Identity Property)

Theorem

Example 1 Solve the following DE by power series

Example 1 Solve the following DE by power series

Expectation

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.)