Math Problem Statement
(-5+x)y'' + (1+4x)y' +3y = 0, y(0) = 3, y'(0)=4
Assuming the solution can be represented by a power series
y=\sum_{n=0}^{\infty}a_nx^n
y' = \sum_{n=1}^{\infty}na_nx^(n-1)
y'' = \sum_{n=2}^{\infty}n(n-1)a_nx^(n-2)
\sum_{n=2}^{\infty}\left(-5+x\right)n\left(n-1\right)a_nx^{n-2} + \sum_{n=1}^{\infty}na_nx^(n-1)+4na_nx^2 + \sum_{n=0}^{\infty}-3a_nx^n = 0
\sum_{n=0}^{\infty} -5(n+2)(n+1)a_(n+2)+(n+1)^2a_(n+1)+(4n+3)a_n x^n=0
a_(n+2) = ((n+1)^2)/(5(n+2)(n+1)) a_(n+1) + (4n+3)/(5(n+2)(n+1)) a_n
Using the initial values and the recursive formula, determine the first few terms of the series solution
y=a_0+a_1x+a_2x^2+a_3x^3+a_4x^4+...
where
a_0 = ?
a_1 = ?
a_2 =?
a_3=?
a_4=?
Solution
We are tasked with computing the first few terms of the series solution for using the initial conditions and the recursive formula.
Given:
The recursive formula is:
Initial conditions:
Step 1: Compute
Using the recursive formula with :
Simplify:
Substitute and :
Thus:
Step 2: Compute
Using the recursive formula with :
Simplify:
Substitute and :
Thus:
Step 3: Compute
Using the recursive formula with :
Simplify:
Substitute and :
Thus:
Final Series Solution (First Few Terms):
The solution is:
Summary of Coefficients:
Would you like to compute additional terms or analyze the behavior of this series further?
Related Questions:
- How does the solution behave for large values of ?
- What is the radius of convergence for this series solution?
- Can the solution be approximated by a simpler polynomial for small ?
- How do the initial conditions affect the solution coefficients?
- Can we verify the solution by substituting it back into the original differential equation?
Tip:
When working with power series, it's helpful to keep terms unrounded until the very end of your calculations to ensure maximum accuracy in intermediate computations.
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Math Problem Analysis
Mathematical Concepts
Differential Equations
Power Series Solutions
Recursive Formulas
Formulas
y = ∑(n=0 to ∞) anx^n
y' = ∑(n=1 to ∞) nanx^(n-1)
y'' = ∑(n=2 to ∞) n(n-1)anx^(n-2)
an+2 = ((n+1)^2 / (5(n+2)(n+1))) an+1 + ((4n+3) / (5(n+2)(n+1))) an
Theorems
Existence and Uniqueness Theorem for Differential Equations
Convergence of Power Series
Suitable Grade Level
Undergraduate Mathematics
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