So you're using a DVI cable (not an HDMI cable plus adapter) to go from the DVI output of your vid card to the DVI input of the TV. Right?
I have an older Samsung TV with similar connections and it works this way. You do have to be careful that the DVI input MUST go to the correct input connector on the back of the TV, whether you're using a true DVI cable or an adapter and HDMI cable. On my TV, I note also that, when using such a connection which does NOT contain any audio signal, you have to provide a separate cable connection to get audio from the computer's green speaker jack to the CORRECT pair of RCA audio inputs on the TV that is associated with that particular DVI / HDMI1 input. I have used mine both via DVI and separate audio cables, and by HDMI cable that required an adapter to "convert" my vid card's DVI output connector to an HDMI output, and they both work. That is, IF I got the correct adapter that passes audio from the vid card out onto the HDMI cable, AND I set Windows to use the audio chip on the vid card (NOT the chip on my mobo) to be my default audio output device.
If it is not working for you, try going through the TV's menu system to try out all of the possible inputs, just in case the labels on the screen are confusing. By the way, on my TV I get to "name" the various input ports to help me keep them straight, so now my Source list includes TV, Comp, Xbox, DVD and VCR.
There might be another issue. On many systems, Windows needs a driver for the video output card, of course. BUT it also needs a driver for the display device being fed by that vid card. Among other things, that display driver interacts with the vid card to ensure you only send out signals the display can handle. Now, you are using your TV, and NOT the display device your cousin had. So maybe you need to remove the driver for your cousin's old display unit (that you don't have) and install a driver for a simple DVI or HDMI device.