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Consider the differential equation x2y'' − 8xy' + 18y = 0; x3, x6, (0, [infinity]). Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. The functions satisfy the differential equation and are linearly independent since W(x3, x6) = Incorrect: Your answer is incorrect. ≠ 0 for 0 < x < [infinity].

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zubam2002

Answer: The equation is differentiated implcitly, with respect to x and y

Step-by-step explanation: x2y - 8xy + 18y = 0

By implicit differentiation: 2y + 2xdy/dx - (8y + 8dy/dx) - 18dy/dx = 0

2y - 8y - 8dy/dx - 18dy/dx +  2xdy/dx = 0

-6y + 2xdy/dx - 26dy/dx = 0

2xdy/dx - 26dy/dx = 6y

dy/dx(2x - 26) = 6y

∴ dy/dx = 6y/2(x - 13) = 3y/(x - 13)

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