Probability and complex function: Unit V: Ordinary Differential Equations

(e) problems based on R.H.S = xn

Solved Example Problems | Ordinary Differential Equations

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Probability and complex function: Unit V: Differential equations: 1. (e) problems based on R.H.S = xn : Examples

1. (e) PROBLEMS BASED ON  R.H.S = xn

Note: The following formulae are important.

(1 + x)-1 = 1 – x + x2 – x3 + …

(1 + x)-1 = 1 + x + x2 + x3 + …

(1 + x)-2 = 1 – 2x + 3x2 – 4x3 + …

(1 + x)-2 = 1 – 2x + 3x2 + 4x3 + …

 

Example 5.1.30. Solve d2y / dx2 - 5 dy/dx + 6y = x2 + 3.

Solution :

Given : d2y / dx2 - 5 dy/dx + 6y = x2 + 3.

i.e., (D2 - 5D+6) y = x2 + 3

The auxiliary equation is m2 - 5m + 6 = 0

⇒ m = 3, m = 2


 

Example 5.1.31 Solve (D2 - D) y = x

Solution:

Given (D2 - D) y = x

The auxiliary equation is m2 – 1 = 0

m2 = 1

m = ± 1


 

Example 5.1.31 Solve (D2 - D) y = x  i.e, d2y / dx2 – dy / dx = x

Solution:

Given (D2 - D) y = x

The auxiliary equation is m2 – m = 0

m(m – 1) = 0

⇒ m = 0, m = 1


 

Example 5.1.33. Solve (D3 - 3D2 - 6D+8) y = x

Solution: Given: (D3 - 3D2 - 6D+8) y = x

The auxiliary equation is (m3 - 3m2 - 6m+8) = 0

⇒ m = 1, m = -2, m = 4


 

Example 5.1.34. Solve [D4 - 2D3 + D2] y = x3.

Solution:

 Given: [D4 - 2D3 + D2] y = x3

The auxiliary equation is m4 - 2m3 + m2 = 0

m2 (m2 - 2m + 1) = 0

m2 (m - 2m + 1) = 0

m2 (m-1)2 = 0

m = 0, 0, m = 1, 1


 

Example 5.1.35. Solve D2 (D2 + 4) y = 96x2

Solution: Given: D2 (D2 + 4) y = 96x2

The auxiliary equation is m2 (m2 + 4) = 0

m2 = 0, m2 + 4 = 0,

m = 0, 0 m = ± 2i


 

Example 5.1.36. Solve (D3 +8) y = x4 + 2x + 1

Solution: Given: (D3 +8) y = x4 + 2x + 1

The auxiliary equation is m3 + 8 = 0


 

Probability and complex function: Unit V: Ordinary Differential Equations : Tag: : Solved Example Problems | Ordinary Differential Equations - (e) problems based on R.H.S = xn

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