Linear Algebra and Differential Equations Chapter 4 Power Se

Linear Algebra and Differential Equations

Chapter 4. Power Series & Numerical Solutions

4.1 The Power Series Method

Please solve 3 questions.

Use power series to solve the differential equation. Give at least six terms explicitly.

Use the power series method to solve the initial-value problem.

Solution

Given that

y = a₀+a₁x+a₂x²+....

we have

y\'\'+ y\'- y = 3x²   =>(1)

let y = a₀+a₁x+a₂x²+a₃x³+a₄x⁴

   y\' = a₁+2a₂x+3a₃x²+4a₄x³

   y\'\' = 2a₂+6a₃x+12a₄x²

sustitude this values in (1) we will get

=> 2a₂+6a₃x+12a₄x²+a₁+2a₂x+3a₃x²+4a₄x³-a₀-a₁x-a₂x²-a₃x³-a₄x⁴ = 3x²

=> -a₀+a₁+2a₂+x(6a₃+2a₂-a₁)+x²(12a₄+3a₃-a₂)+x³(4a₄-a₃)-a₄x⁴ = 3x²

comparing the coefficients of eq both sides we will get

-a₀+a₁+2a₂ = 0

6a₃+2a₂-a₁ = 0

12a₄+3a₃-a₂ =3

4a₄-a₃ = 0

a₄ =0

by solving above equations we will get

a₀ = -12, a₁ = -6, a₂ = -3, a₃ = 0, a₄ = 0

then the eq of y is

y= a₀+a₁x+a₂x²+a₃x³+a₄x⁴

y= -12-6x-3x²+0x³+0x⁴

y = -12 - 6x -3x²