# GATE1998(1): Higher Order Differential Equation

### 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