There are various ways to solve this problem. One of them is to replace the faulty switch with a new one. But then again, there is a possibility that the new switch will also go bad with time. Another solution is to insert pins into the power button, so that it is kept pressed all the time. This method requires patience and there is no guarantee that it will work every time. So, a permanent solution would be to take the switch out of the circuit. It means we need to short the switch permanently. This is a very easy trick, all you need is a screwdriver and a soldering iron.
Step 1: Open the Modem
The red arrows show the position of latches. After opening the modem, we need to do our real task, short the terminals of the power button. Here is the inside look of modem.
Step 2: Soldering the Switch
If you have done everything right, your work is over. Before you put all the things back together, its better to test the modem. Just plug in the power supply to the modem and it should switch on right away without the need to press the power button. If its working correctly, put all the things back together. Also, you have the opportunity to clean the insides of the modem. Its better to use compressed air can, but if you don't have one you can gently wipe it with a dry cloth.
This tutorial can also be found at www.electronicfunda.blogspot.in
Now enjoy your broadband without worrying about the power switch.