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].

Question

Mathematics

Kassandra Odonnell

29 July, 11:00

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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Nathalia · Answer 1

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)