### Question: GATE2012 (2 Marks)

Consider the differential equation \(x^2 (d^2y/dx^2) + x(dy/dx) – 4y = 0 \) with the boundary conditions of y(0) = 0 and y(1) = 1. The complete solution of the differential equation is

\((A) x^2 \) | \(\displaystyle (B) sin \bigg (\frac{\pi x}{2}\bigg) \) |

\(\displaystyle (C) e^x sin \bigg (\frac{\pi x}{2}\bigg) \) | \(\displaystyle (D) e^{-x} sin \bigg (\frac{\pi x}{2}\bigg) \) |

### Solution:

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^{2} – 4) = 0

(t + 2)(t – 2) = 0

Complementary Function: y = C_{1}x^{ –2 }+ C_{2}x^{2}

Applying Boundary Conditions:

y(0) = 0

0 = C_{1}/0 + C_{2} x 0; which means that C_{1 }= 0

y(1) = 1

1 = C_{2} x 1

C_{2} = 1

Solution: y = x^{2}