GATE1998(1): Higher Order Differential Equation

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Question: GATE1998 (2 Marks)

The general solution of the differential equation \(\displaystyle x^2 \frac{d^2y}{dx^2} – x \frac{dy}{dx} + y = 0 \) is

(a) Ax + Bx2  (A,B are constants)                   (b) Ax + B logx    (A,B are constants)
(c) Ax + Bx2 logx   (A,B are constants) (d) Ax +Bx log x   (A, B are constants)

Solution:

x2y’’ – xy’ + y = 0

This is a second order Euler-Cauchy Differential Equation

Take y = xt

y’ = txt-1

y’’ = t(t-1)xt-2

The differential equation becomes xt (t – 1)2 = 0

(t – 1)2 = 0

Complementary Function: y = Ax + Bxlogx

Answer: (D)