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If, A and B are arbitrary constants then what will be the differential equation of y = Ax + B/x?(a) x^2 d^2 y/dx^2 – xdy/dx + y = 0(b) x^2 d^2 y/dx^2 + xdy/dx + y = 0(c) x^2 d^2 y/dx^2 + xdy/dx – y = 0(d) x^2 d^2 y/dx^2 – xdy/dx – y = 0I have been asked this question in quiz.My doubt is from Linear Second Order Differential Equations topic in section Differential Equations of Mathematics – Class 12

Answer»

Right answer is (c) X^2 d^2 y/dx^2 + xdy/dx – y = 0

Easiest EXPLANATION: GIVEN, y = Ax + B/x

=> xy = Ax^2 + B……….(1)

Differentiating (1) with respect to x, we get,

d(xy)/dx = d/dx(Ax^2 + B)

or, xdy/dx + y = A * 2x……….(2)

Differentiating again with respect to x, we get,

x*d^2y/dx^2 + dy/dx + dy/dx = A*2……….(3)

Eliminating A from (2) and (3) we get,

x^2 d^2 y/dx^2 + 2xdy/dx = 2Ax[multiplying (3) by x]

or, x^2 d^2 y/dx^2 + 2xdy/dx = xdy/dx + y [using (2)]

or, x^2 d^2 y/dx^2 + xdy/dx – y = 0


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