Verify that the given functions y1 and y2 form a fundamental set of solutions of

Question

Verify that the given functions y1 and y2 form a fundamental set of solutions of the reduced equation of the given nonhomogeneous equation; then find a particular solution of the nonhomogeneous equation and give the general solution of the equation. y" - 2/x2 y = 3 - x-2; y1(x) = x2, y2(x) = x-1.

Explanation / Answer

To verify the fundamental solution , find the solution of y'' - (2/x^2)y = 0 Put x^2 in the above equation => LHS = 2 - (2/x^2)x^2 =0 Put x^-1 in the above equation => LHS = 2/x^3 - (2/x^2)x^-1 =0 Hence both are fundamental solution of the given equation. y0 = C1 x^2 + C2 x^-1 Now using Lagrange's Method, C1' x^2 + C2' x^-1 = 0 C1' (2x) + C2' (-1/x^2) = 3 - x^-2 C1' (2x + x ) = 3 - x^-2 => C1' = x^-1 - (1/(3x^3)) => C1 = lnx + (1/(6x^2)) + A1 C2' = (1/3) - x^2 => C2 = (1/3)x - (x^3/3) + A2 Particular solution = lnx * x^2 + (1/6) + (1/3) - (x^2/3) [(as PS = C1 x^2 + C2 x^-1)] General Solution = A1 x^2 + A2 x^-1 + Particular solution

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