UPSC 2018 Maths Optional Paper 1 Q5d-i — Step-by-Step Solution

5 marks · Section B

Laplace transform · ODEs · asked 2× in 13 yrs · Read the full method →

Question

Find the Laplace transform of $f(t)=\dfrac{1}{\sqrt t}$ .

$\mathcal L\{t^n\}=\Gamma(n+1)/s^{n+1}$ for $n>-1$ , with $\Gamma(1/2)=\sqrt\pi$ ; or direct reduction to the Gaussian integral.

For $n>-1$ the standard transform is

\mathcal L\{t^{n}\}=\int_0^\infty e^{-st}t^{n}\,dt=\frac{\Gamma(n+1)}{s^{\,n+1}},\qquad s>0.

Here $f(t)=t^{-1/2}$ , so $n=-\tfrac12$ (and $n>-1$ , so the integral converges at $t=0$ ):

\mathcal L\{t^{-1/2}\}=\frac{\Gamma\!\left(\tfrac12\right)}{s^{\,1/2}}.

Using $\Gamma\!\left(\tfrac12\right)=\sqrt\pi$ :

\mathcal L\!\left\{\frac1{\sqrt t}\right\}=\frac{\sqrt\pi}{\sqrt s}.

\boxed{\;\mathcal L\!\left\{\frac{1}{\sqrt t}\right\}=\sqrt{\frac{\pi}{s}},\qquad s>0.\;}