Question

(D²-1) y
= 2+5x. find particular integral
​

Answer 1

The correct answer is: y = C₁eˣ + C₂e⁻ˣ – 2 – 5x

Given: (D² – 1) y = 2 + 5x.

To find: Particular Integral.

Solution:

(D² – 1) y = 2 + 5x

⇒ y = (2 + 5x)(D² – 1)

[m² – 1 = 0, m = 1, -1]

Complimentary Function (C.F.) is C₁eˣ + C₂e⁻ˣ

Now,

Particular Integral (P.I.) is

y = 1/(D² – 1) × (2 + 5x)

  = – 1/(1 – D²) × (2 + 5x)

  = – 1/(1 + D)(1 – D) × (2 + 5x)

  = – ½ × [(1 + D) + (1 – D)]/[ (1 + D)(1 – D)] × (2 + 5x)

  = – ½ × [1/(1 – D) × (2 + 5x) + 1/(1 + D) ×(2 + 5x)]

  = – ½ × [(1 – D)⁻¹ × (2 + 5x) + (1 + D)⁻¹ × (2 + 5x)]

  = – ½ × [(1 + D + D² +…)(2 + 5x) + (1 – D…)(2 + 5x)]

  = – ½ × [2 + 5x + 5 + 2 + 5x – 5]

  = – ½ × [4 + 10x]

  = – 2 – 5x

∴ y = – 2 – 5x

Therefore,

General Solution is  y = C₁eˣ + C₂e⁻ˣ – 2 – 5x

Thus, the correct answer is: y = C₁eˣ + C₂e⁻ˣ – 2 – 5x

Additional Information:

• When y = f(x) + cg(x) is the solution of an ODE (Ordinary Differential Equation), ‘f’ is known as the Particular Integral (P.I.) and ‘g’ is known as the Complementary Function (C.F.).

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