APM2616
ASSIGNMENT 04
DUE 2025
, Question 1: [25 Marks]
Problem Statement
We are given the second-order linear differential equation:
x2 y ′′ (x) + x2 y ′ (x) + xy(x) = 0,
with the initial conditions:
y(1) = 1, y ′ (1) = 1.
Our goal is to solve the equation and plot the function y(x) over the interval [1, 50].
Step 1: Simplify the Differential Equation
We start by simplifying the equation:
x2 y ′′ + x2 y ′ + xy = 0.
Factor out an x from all terms:
x (xy ′′ + xy ′ + y) = 0.
Since x ̸= 0 in the domain of interest, divide through by x:
xy ′′ + xy ′ + y = 0.
Step 2: Convert to First-Order System
Let:
y1 = y(x), y2 = y ′ (x).
Then the derivatives become:
y1′ = y2 , y2′ = y ′′ .
1
ASSIGNMENT 04
DUE 2025
, Question 1: [25 Marks]
Problem Statement
We are given the second-order linear differential equation:
x2 y ′′ (x) + x2 y ′ (x) + xy(x) = 0,
with the initial conditions:
y(1) = 1, y ′ (1) = 1.
Our goal is to solve the equation and plot the function y(x) over the interval [1, 50].
Step 1: Simplify the Differential Equation
We start by simplifying the equation:
x2 y ′′ + x2 y ′ + xy = 0.
Factor out an x from all terms:
x (xy ′′ + xy ′ + y) = 0.
Since x ̸= 0 in the domain of interest, divide through by x:
xy ′′ + xy ′ + y = 0.
Step 2: Convert to First-Order System
Let:
y1 = y(x), y2 = y ′ (x).
Then the derivatives become:
y1′ = y2 , y2′ = y ′′ .
1
