GATE (TF) Textile 2021 Question Paper Solution | GATE/2021/TF/03

Question 03 (Textile Technology & Fibre Science)

The smallest positive real number \lambda, for which the following problem

    \[y"(x) + \lambda y(x) = 0\]

    \[y'(0) = 0, \quad \quad y(1) = 0\]

has non-zero solution, is

Answer / Solution

    \[y"(x) + \lambda y(x) = 0\]

Taking Laplace both side-

[s^2 \times \bar{y(0)}-s\times y(0)-{y(0)}']+\lambda \times \bar{y(0)}=0

(s^2+\lambda) \times \bar{y(0)}-s\times y(0)-{y(0)}'=0

(s^2+\lambda) \times \bar{y(0)}-s\times y(0)-0=0

let, \bar{y(0)}=k

(s^2+\lambda) \times \bar{y(0)}-s\times k=0

\bar{y(0)}=\frac{s\times k}{s^2+\lambda}

Taking inverse laplace both side

L^-1 y(0)=kL^-1 \frac{s}{s^2+\lambda}

y=kcos\sqrt \lambda x



0=kcos\sqrt \lambda

cos\sqrt \lambda=0

cos\sqrt \lambda=cos\frac{\pi}{2} ,cos\frac{3\pi}{2} ,cos\frac{5\pi}{2}

\sqrt \lambda=\frac{\pi}{2}

\lambda=\frac{\pi^2}{4} (Ans)

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