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

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

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$y"(x) + \lambda y(x) = 0$

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

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

$y=kcos\sqrt \lambda x$

Given-

y(1)=0

$0=kcos\sqrt \lambda$

$cos\sqrt \lambda=0$

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

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

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