GATE2012: Higher Order Differential Equation

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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 = xt

y’ = txt-1

y’’ = t(t-1)xt-2

The differential equation becomes xt (t2 – 4) = 0

(t + 2)(t – 2) = 0

Complementary Function: y = C1x –2 + C2x2

Applying Boundary Conditions:

y(0) = 0

0 = C1/0 + C2 x 0; which means that C1 = 0

y(1) = 1

1 = C2 x 1

C2 = 1

Solution: y = x2

Answer: (A)