APM2611 Assignment 4 DUE 24 September 2025

APM2611
Assignment 4
DUE 24 September 2025

, APM2611

Assignment 4

DUE 24 September 2025



Question 1: Solving the Initial Value Problem Using the Power Series Method

Problem Statement

Solve the initial value problem:

xy′′ + 4 y′ = 2 , y(0) = 0 , y′(0) = 1 ,

using the power series method.

Solution Approach

The power series method assumes a solution of the form .

This is suitable for differential equations with variable coefficients. The method involves
substituting the series into the ODE, equating coefficients, and solving for an.

Step-by-Step Solution

Compute the derivatives:

.




Multiply y′′ by x:


.

Adjust the index:

∞

xy′′ = X( n+ 1) nan+1 xn.
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