hello, i desperately need help with differential equations- newton's law of cooling
i need to solve this D.E.........
dT/dt= -k(T-8)+H where k is a positive constant and h=1........
when t=0 T=16 PLEASE HELP ME!!!!!!!!!!!!!!!!!!!
hello, i desperately need help with differential equations- newton's law of cooling
i need to solve this D.E.........
dT/dt= -k(T-8)+H where k is a positive constant and h=1........
when t=0 T=16 PLEASE HELP ME!!!!!!!!!!!!!!!!!!!
$\displaystyle \frac{dT}{dt} = -k(T - 8) + H$
$\displaystyle \frac{dT}{dt} + kT = 8k + H$
The homogeneous equation is
$\displaystyle \frac{dT}{dt} + kT = 0$
and has a solution
$\displaystyle T_h(t) = Ae^{-kt}$
The particular solution will be of the form
$\displaystyle T_p(t) = B$
since the RHS of the differential equation is simply a constant. You can plug this into the original differential equation to get a value for B in terms of k and H.
Thus the general solution of this equation will be:
$\displaystyle T(t) = Ae^{-kt} + B$
where A is a constant to be fit to your initial conditions.
-Dan