To travel $708\text{km}$, an Express train takes $6\text{ hours}$ more than Rajdhani train. If however, the speed of the Express train is doubled, it takes $3\text{ hours}$ less than Rajdhani train. What is the speed of Rajdhani train?

Let the speed of the Express train be $x\text{km/hr}$ and the speed of the Rajdhani train be $y\text{km/hr}$.From the first condition:
$$\frac{708}{x}-\frac{708}{y}=6⟹\frac{1}{x}-\frac{1}{y}=\frac{6}{708}=\frac{1}{118}\text{— (Equation 1)}$$
From the second condition (speed of Express is doubled):
$$\frac{708}{y}-\frac{708}{2x}=3⟹\frac{1}{y}-\frac{1}{2x}=\frac{3}{708}=\frac{1}{236}\text{— (Equation 2)}$$
Add Equation 1 and Equation 2:
$$\left(\frac{1}{x}-\frac{1}{y}\right)+\left(\frac{1}{y}-\frac{1}{2x}\right)=\frac{1}{118}+\frac{1}{236}$$
$$\frac{1}{2x}=\frac{2+1}{236}=\frac{3}{236}$$
$$\frac{1}{x}=\frac{6}{236}=\frac{3}{118}$$
Substitute $\frac{1}{x}=\frac{3}{118}$ into Equation 1:
$$\frac{3}{118}-\frac{1}{y}=\frac{1}{118}$$
$$\frac{1}{y}=\frac{3}{118}-\frac{1}{118}=\frac{2}{118}=\frac{1}{59}$$
$$y=59\text{km/hr}$$
Thus, the speed of the Rajdhani train is $59\text{km/hr}$.