1 Compute the Wronski determinant of the given functions Are

(1) Compute the Wronski determinant of the given functions. Are these functions linearly

independent or linearly dependent?

(i) y1(x) = 2x 3, y2(x) = x^2 + 1, y3(x) = 2x^2 x;

(ii) y1(x) = 2x 3, y2(x) = 2x^2 + 1, y3(x) = 3x^2 + x;

(iii) y1(x) = 2x 3, y2(x) = x^2 + 1, y3(x) = 2x^2 x, y4(x) = x^2 + x + 1.

(2) Find the general solution of

(i) y^(4) 8y = 0;

(ii) x^2y 2xy + 3y = 0;

(iii) x^2y 3xy + 4y = 4 ln x (x > 0);

Solution

2. i) (y\')⁴ -8y\' = 0

Auxiliary equation is given by:

D⁴ - 8D = 0

D(D³ -8) = 0

D= 0 or (D³ -8) = 0

D= 0 or D= 2,2,2

The general solution is given by:

y = C₁e^0x + C₂e^2x + C₃xe^2x + C₄x²e^2x

y = C₁ + (C₂ + C₃x + C₄x²)e^2x

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