English: So what is that this means this cannot positive that means energy has to be zero that means w has to be zero because implies []because this is integrant is positive w(x,t) = 0 for every t. So this means w(x,t) = u₁(x,t) − u₂(x,t) = 0 that is nothing but u₁(x,t) = u₂(x,t) so that shows that if you have two solutions of the initial value problem u₁(x,t) and u₂(x,t) they are actually same, okay you take the difference use this energy argument and show that the uniqueness of the initial value problem that we have considered, okay.
Kannada: ಇದರರ್ಥ ಶಕ್ತಿ ಶೂನ್ಯವಾಗಿರಬೇಕು, ಅಂದರೆ  w ಶೂನ್ಯವಾಗಿರಬೇಕು ಏಕೆಂದರೆ [] ಮತ್ತು ಇದು ಸಮಗ್ರವಾಗಿರುವ ಧನಾತ್ಮಕವಾಗಿದೆ, ಪ್ರತಿ t ಗೆ w(x,t) = 0. ಇದರರ್ಥ w(x,t) = u₁(x,t) − u₂(x,t) = 0, ಅದು u₁(x,t) = u₂(x,t) ಅಲ್ಲದೆ ಬೇರೆ ಏನೂ ಅಲ್ಲ.