# Thread: urgent help needed with differential equations

1. ## urgent help needed with differential equations

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........

2. Originally Posted by passarez
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........
$\displaystyle \frac{dT}{dt} = -k(T - 8) + H$

$\displaystyle \frac{dT}{dt} + kT = 8k + H$

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