Answer to Question #104078 in Differential Equations for mm

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.