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**HallsofIvy** No, we **don't** "see that at u(1,0) , u(2,0 ) and u(3,0) , the t = 10 degree". The two ends are held at 0 and 10 degrees [b]until a steady state temperature is achieved." "Steady state" means that as long as conditions are maintained (the bar is insulated and the end A, at x= 0 is 0, the end at B, x= 4, is at 10) the temperature will not change- the given d.e. is satisfied by a function in x that does not depend on t.

That means that $\displaystyle \frac{\partial^2 u}{\partial t^2}$ and so, for all x, between 0 and 4, $\displaystyle \frac{\partial^2 u}{\partial t^2}$ must also be equal to 0 in order that the differential equation is satisfied. Since the second derivative is 0, the first derivative is a constant and the function itself is linear: u(x, 0)= ax+ b. So u(0, 0)= b= 0 and u(4, 0)= 4a= 10. a= 10/4= 5/2. u(x, 0)= (5/2)x.