Applying $L.H.$ rule
$L = \mathop {\lim }\limits_{t \to x} \begin{array}{*{20}{c}}
{2t\,f\left( x \right) - {x^2}f'\left( t \right)}\\
1
\end{array} = 1$
$2t\,f\left( x \right) - {x^2}f'\left( x \right) = 1$
solving above differential equation, we get
$f\left( x \right) = \frac{2}{3}{x^2} + \frac{1}{{3x}}$
Put $x = \frac{3}{2}$
$f\left( {\frac{3}{2}} \right) = \frac{2}{3} \times {\left( {\frac{3}{2}} \right)^2} + \frac{1}{3} \times \frac{2}{3}$
$ = \frac{3}{2} + \frac{2}{9} = \frac{{27 + 4}}{{18}} = \frac{{31}}{{18}}$
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