Question

Newton’s law of cooling assumes that air at room temperature is blown past the cooling body (forced cooling). For cooling in still air (natural cooling) a better modal is to assume that the rate of temperature decrease of the cooling body is directly proportional to the the (5/4)th power of the difference between the temperature u of the body and the temperature s of the surrounding air.
i) Write the law for natural cooling as a differential equation. Is this equation linear?
ii) Solve the equation obtained in i) above assuming that initially, the temperature of the cooling
body was u0.

Answer 1

$-du/dt=k(u-s)^{5/4}$

is the differential equation for law of natural cooling.

Now, as degree of the differential equation is 1, thus it is linear.

$\int_{u_0}^u du/(u-s)^{5/4}=-\int_0^t k dt$

$(u-s)^{-1/4} -(u_0 -s)^{-1/4}=-4kt$

$u=s+((u_0 -s)^{-1/4}-4kt)^{-4}$

is the solution for the above differential equation.