APM2611 Assignment
4 2025 (Answer
Guide) - Due 24
September 2025
QUESTIONS WITH 100%
VERIFIED AND
CERTIFIED ANSWERS.
, 1|Page
APM2611 Assignment 4 2025 (Answer Guide) - Due 24 September 2025
VERIFIED AND CERTIFIED ANSWERS. WRITTEN IN REQUIRED FORMAT AND
WITHIN GIVEN GUIDELINES. IT IS GOOD TO USE AS A GUIDE AND FOR
REFERENCE, NEVER PLAGARIZE. Thank you and success in your academics.
UNISA, 2025
APM2611 Assignment 4 - Question 1
Question 1
Use the power-series method to solve the initial-value problem:
y'' - x y' + 4y = 2, y(0)=0, y'(0)=1.
Solution
1. Assume a power series
Assume y(x) = Σ (from n=0 to ∞) aₙ xⁿ.
Then
y'(x) = Σ (from n=0 to ∞) (n+1)aₙ₊₁ xⁿ,
y''(x) = Σ (from n=0 to ∞) (n+2)(n+1)aₙ₊₂ xⁿ.
Also, -x y'(x) = -Σ (from n=0 to ∞) n aₙ xⁿ.
Substituting into the differential equation gives:
Σ (n=0 to ∞)[(n+2)(n+1)aₙ₊₂ + (4-n)aₙ] xⁿ = 2.
2. Recurrence relations
For n=0:
2a₂ + 4a₀ = 2.
For n ≥ 1:
(n+2)(n+1)aₙ₊₂ + (4-n)aₙ = 0
⇒ aₙ₊₂ = -(4-n)/( (n+2)(n+1) ) * aₙ.
4 2025 (Answer
Guide) - Due 24
September 2025
QUESTIONS WITH 100%
VERIFIED AND
CERTIFIED ANSWERS.
, 1|Page
APM2611 Assignment 4 2025 (Answer Guide) - Due 24 September 2025
VERIFIED AND CERTIFIED ANSWERS. WRITTEN IN REQUIRED FORMAT AND
WITHIN GIVEN GUIDELINES. IT IS GOOD TO USE AS A GUIDE AND FOR
REFERENCE, NEVER PLAGARIZE. Thank you and success in your academics.
UNISA, 2025
APM2611 Assignment 4 - Question 1
Question 1
Use the power-series method to solve the initial-value problem:
y'' - x y' + 4y = 2, y(0)=0, y'(0)=1.
Solution
1. Assume a power series
Assume y(x) = Σ (from n=0 to ∞) aₙ xⁿ.
Then
y'(x) = Σ (from n=0 to ∞) (n+1)aₙ₊₁ xⁿ,
y''(x) = Σ (from n=0 to ∞) (n+2)(n+1)aₙ₊₂ xⁿ.
Also, -x y'(x) = -Σ (from n=0 to ∞) n aₙ xⁿ.
Substituting into the differential equation gives:
Σ (n=0 to ∞)[(n+2)(n+1)aₙ₊₂ + (4-n)aₙ] xⁿ = 2.
2. Recurrence relations
For n=0:
2a₂ + 4a₀ = 2.
For n ≥ 1:
(n+2)(n+1)aₙ₊₂ + (4-n)aₙ = 0
⇒ aₙ₊₂ = -(4-n)/( (n+2)(n+1) ) * aₙ.
