### 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 + Bx^{2} (A,B are constants) |
(b) Ax + B logx (A,B are constants) |

(c) Ax + Bx^{2} logx (A,B are constants) |
(d) Ax +Bx log x (A, B are constants) |

### Solution:

x^{2}y’’ – xy’ + y = 0

This is a second order Euler-Cauchy Differential Equation

Take y = x^{t}

y’ = tx^{t-1}

y’’ = t(t-1)x^{t-2}

The differential equation becomes x^{t }(t – 1)^{2} = 0

(t – 1)^{2} = 0

Complementary Function: y = Ax + Bxlogx