Chapter 1 - First-Order Differential Equations - 1.6 First-Order Linear Differential Equations - Problems - Page 59: 24

$T=ce^{-kt}+T_0$

Newton’s law of cooling: $\frac{dT}{dt}=k(T_0-T)$ $\frac{dT}{dt}+kT=kT_0$ where $k$ is proportionality constant and $T_0$ is the environment's temperature Intergrating factor: $I=e^{\int kdt}=e^{kt}$ Multiply both sides by the intergrating factor: $T=I^{-1}(c_1+\int IkT_0)$ C is constant of integration $T=e^{-kt}(c_1+kT_0\int e^{kt}dt)$ $T=ce^{-kt}+T_0$ The solution is $T=ce^{-kt}+T_0$