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The respective expressions for complimentary function and particular integral part of the solution of the differential equation d⁴y/dx⁴ + 3 d²y/dx² = 108x² are

1. [c1 + c2x + c3 sin √3x + c4 cos √3x] and [3x⁴ - 12x² + c]
2. [c2x + c3 sin √3x + c4 cos √3x] and [5x⁴ - 12x² + c]
3. [c1 + c3 sin √3x + c4 cos √3x] and [3x⁴ - 12x² + c]
4. [c1 + c2x + c3 sin √3x + c4 cos √3x] and [5x⁴ - 12x² + c]

Correct answer: [c1 + c2x + c3 sin √3x + c4 cos √3x] and [3x⁴ - 12x² + c]

Solution

The correct option includes the general solution of the associated homogeneous equation, which consists of constant terms, linear terms, and sinusoidal functions, along with the particular integral that correctly addresses the non-homogeneous part of the differential equation, resulting in a polynomial expression that matches the degree of the driving function.

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